\r\,

It -2'15

AN TNVESTIGATION OF THE OAME

OF POKER BY COMPUTER BASED ANALYSIS

by

A. Rlstlcz B.Sc. Hons. (Adelalde)

A thesls submltted for the degree ofDoctor of Phllosophy

1n the Unlverslty of AdelaldeDecember, L973.

SUMMARY

The researeh reported ln thls thesls has been under-

taken wlth the obJectlve of flndlng optlmal strategles forpoker uslng both slmulatlon and game-theoretlc methods.

Ïn the lntroductlon the nature of the problem 1s presented

and the maln work already done in thls area ls revlewed,

Next the simulatlon of poker 1s consldered and the

computatlonal difflcultles of thls approach are lnvestlgated

1n some detalI. It 1s concluded 1n thls thesls that an

analyt1c,/numerlcal approach offers better prospects of

success than slmulatlon. Accordlngly the numerleal approach

to poker ls theoretlcally formulated and new methods of

solutlon are developed whlch are slgnlflcantly faster than

exlstlng methods found ln the llterature.

These methods are then used to solve 2, 3 and 4-

person poker-like games. In the course of thls work a new

lntegral relevant to the solutlon of poker-llke games 1s

evaluated. Slmulatlon methods are used to check the work

wherever posslble.

The results obtaÍne<i. from these solutions o.re unique fortwo reasons. First, even though simplifying assÌrtptions were mad.e

i-n many phases of this work, the games solved are sufficientlyrealistj.c to be compared wlth a commonly played. variety of poker

and the soLutions are shown to agree cIose1y, 1n most d"etail-sr with

the strategles used by experlenced playersr even though some of the

results are in less than coinplete a,greement. Second.Iy, this appears

to be the(1)

first tlme that a sorut-lon has been obt.inetl for a-lr-pensonpoker-llke game.

The study then reports the results of the appllca-tlon of thls wonk to two practlcal problems not dlrectlyconnected wlth poker. The flrst of these relates to net-worlts and formulates a new crlterlon of optlmallty of frowwhlch 1s currently the subJect of further research by

another worker. The second is a problem in buslness

management.

The thesls ends wlth a diseusslon 1n which generat

concluslons are drawn from the whole of the work and 1n

whlch neÏ¡ areas of research ane ld.entifled. rn partlcularmore research on numerlcal methods r oû the methodology ofslmulatlon, and ln applled games theory ls recommended.

(rr¡

ACKNOV'ILEDGEMENTS

I would l1ke to express my appnoel_atlr¡n ofthe guldance, encouragement and help given by

Dr. J.t.c. Maeask1l1 during the course of my researeh,and to Professor J. ovenstone for hls advice durlngthe prellmlnary stages of my work. I also thankProfessor F. Hlrst 1n hls capacity as permanent Head

of the Department of Computlng Science for hlsencouragement and suppont, and Dr. r.N. capon, Dlrectorof the unlversity of Aderalde computlng centre, and h1s

staff, for thelr help and tolerance. The wrlter' 1s

lndebted to Professor R.B. potts, Dr. c.E.M. pearce and

Mr. R.J. Harrls for vaLuable dlscusslons and toIUiss J. Potter for metlculous typlng.

( lv)

TABLE OF CONTENTS Page

(1)

(111)

(lv)

B

B

9

10

11

T2

13

I3

13

14

14

Summary

Slgned Statement

AcknowLedgements

Chapter" 1

INTRODUCTTON

1.1 Staternent of problem

I.2 Maln alm of the work of thls theslsREVIEÌ^I OI' RESEA RCH INTO POKER

Methods of elasslfylng poker-I1ke games

Games of hlstorlcal lnterest1.4.1 The games of von Neumann and

MorgensternI.\.2 The games of G1111s et aI. and

of Bellman and Blackwelt

t. 4. 3 Tfre games of Kar1ln and Restreppo,and Prultt

Games of dlrect slgnÍ_flcance to thls thesls1.5.1 Kuhnts 2-person game

1.5.2 Nash and Shapley's J-person game

F'rledman I s game

Numerlcal technlques

Computer slmulatlon of poker

NEI/'I lr,lORK PRESENTED IN IHIS THESIS

I

I

2

3

3

5

5

6

1.3

1.4

1.5

1.6

r.7r.B

I

1.9

1.10

1.11

I.I2

Poker slmulation

A new solutlon method for n-person games

Evaluatlon of a speclal lntegralFormulatlon and solutlon of a new poken-I1ke game

1.13 Applicatlons

LAYOUT OF THESTS

Chapter 2 Slmu1atl on of Poker

INIRODUCTION t7Introduction ITPrlnc1p1-es of poker slmulatlon tTResults achleved by other workers ZL

Factors lnvolved ln settlng up a simulation 2IDETAILED METHOD OF STMULATION

1.14 Layout of thesls

2"t2.2

2.3

2,4

2.5 Rules of the game

2.6 Slmulation of shuffllng2 rT Impnovement of hand

2.8 Calculation of probablllties2.9 Bettlng strateglesDÏSCUSSION

Page

14

I5

15

22

26

28

2g

37

43

4S

46

4B

50

51

54

54

2.10

2 .11

2 .12

2.r3

Checklng of results from computer runs

ïnteractlve poker

fnvestigatlon of poker strategles byslmulatlon

Concludlng remarks

Chapter 3 Lookahead (tAH ) etso rlthm3.1 An Introductlon

3,2 Mathematlcal statement of problem

3.3 Rosenrs method of numerical solutlon3.4 The Lookahead algorlthm

3.5

3.6

3.7

3.8

3.9

The LAH algorlthm 1n a Z-person game

Equlvalence between LAH solublon and e.p.Generallzation of LAH algorlthm ton-person case

Appllcatlons of LAH algorlthm

Comparlson between Rosents modlfled methodand the LAH algorlthm

Page

55

59

6\

66

73

Chapter 4

Formulatlon and solutlon of a 2 -ï)erson Doker-11ke Aame

4.r

\.24.3

4.4

4.5

4.6

4.7

Chapter 5

Introductlon

Strategles of playens

Evaluatlon of payoff functlons and. solutlonAnalytlc solutlon of the game

4.4.f fn1t1aI assumptlons

4.4.2 Showlng that the solutlon found1s an e.p.

4. 4. I The depende.nce of the analytlcsolutlon upon the appnoximatesolutlon

Valldatlon of hand lmprovement slmulatlon

Non-optlma1 solutlons

Discusslon of results

75

8r

B¡

93

93

9B

99

Solutlon of 3 and 4 person Eame (3

5.1 Introduction

5.2 Solutlon of the 3-PG

5.2.I Rules of play

5.2.2 Strategles

100

104

105

108

109

109

109

-PG and 4-Pc)

5.3

5.2-3 Evaluatlon of payoff functlons

5.2.4 So1utlon of the 3-Pc

Solút1on of thê 4-pO

5.3.1 Evaluatlon of payoff functlons

5.3.2 Speclal method to speed solutlonof the 4-PC

5.3.3 Scilutlon of the 4-PO

Checklng the 3-PG and 4-Pc payoff functlons

Dlscusslon of results

5,5.I Comparlson between solutlonsto the 2-PG and 3-PG

5.5.2 Practlca1 Interpretation of thesoLutlon to the 4-PC

5.5,3 Dlscusslon of solutlon to the4-PG

5.5.\ Comparlson of analytlc solutlonwlth the strategies used byexperienced players

Introductlon

Appllcatlon of game theoretlc methodsto a problem ln networks

6.?.1 Theoretlcal background

6.2.2 Game theonetlc netwonk optlmLza-tlon

6.2.3 Dlscusslon of results

Buslness model

6.3.f Deflnltlon of buslness model

6.3.2 Dlscusslon of buslness model

Page

l,L2

tt?IL7

r17

].26

128

128

L32

r32

135

139

143

148

148

148

t52

153

155

r62

163

5.4

5,5

Chapter 6

Appllcatlons of the work carrled out 1n this study

6.r6.2

6.3

Chapter 7

Sumniary. co¡_clusloqs and discussion o f new work

7 .t Sunmary

7.2 Approach to the problem posed Ln thlsthesls

7.3 Further aneas of reseanch

Index of appendlces

Appendlx A: Evaluatlon of Payoff FunctlonIntegral

Random Number Generator

Calculatlon of Probabllltles

Appendlx B:

Appendlx C:

Index of Fleures

Index of Tables

Blbllography

PaÉte

J,65

L67

l.72

TT6

L99

202

209

?to

2t2

1

CHAPTER 1

INTRODUCTION

1, I Statenent of problem

Before maklng a general statement of the problem tobe dealt wlth 1n thts thesls the followlng 2 polnts should

be noted.

(a) The mathematlcal treatment of poker can have pnactlcal

appllcatlons ln areas not dlrectly related to poken.

The followlng quote from Karlln, (19), supports thlsassertlon (also see Prultt (29)):rrThe many nespects 1n which buslness, po]ltlcs and

war resemble poker should be evldent, Hence, any

progress 1n our mathematlcal understanding of poker

games can have lts counterpart lnterpretatlon lnmany nelevant elrcumst¿u1ces of llfe. rl

(b) Poker 1s posslbly the most complex of card games.

Indeed Karlln, (19), states that:rflt ls the consldered oplnlon of many expert card

players thaü poken nequlres the most sklll and depth

of any card game. rl

The lntrlcacles of poker may be appreclated by read-

lng one of the many þooks wrltten about the subJect, forexample ffPoker: Gane of Sklllrr, (30). As a consequence of

thls complexlty the nathematlcal tneatment of poker must be

restrlcted to slmpllfied poker-Ilke games (see the llterature

,.¡

"ñ

2.

survey glven tater ln thls chapter).

In thls study the game of poker has been approached

ln such a way that, although certaln essentlal character-

1st1cs of the game are retalned, slmpllfylng assumptlons

make the problem amenabl-e to mathematlcal solutlon. Num-

erlc methods, suitable for use on the computer are used to

obtaln solutlons whleh are then dlreetly appllcable to a

commonly played varlety of poker.

Vanlous appllcatlons of thls work to other, not

dhectly related areas, are then consldened.

I.2 Maln alm of the work of thls thesls

The maln dlffleulty Ln the mathematical treatment

of poker-llke games arlses from computatlonal dlfflcultles(see later ln thls chapter and chapter 5). Thls worlt

hypotheslses that, by uslng numerlc methods (suitable for

lmplementatlon on a computer), further progress may be made

ln solvlng more reallstlc models than have been herreto

posslble (see chapter 5).

To flnd these solutlons a new mathematlcal technlque

for handllng the partlcular type of games treated here has

been developed (see chapter 3). Although the appllcabllltyof poker to buslness has been suggested by Prultt, (29), and

Karlln, (19), no concrete examples could be found ln the

llterature. Thus 1t was an obJect of thls study to provide

a speclflc example of the appllcatlon of the above work to a

3.

buslness sltuatlon (see chapter 6). Other appllcatlons to

poker simulatlon and the theory of networkb are also glven

in chapters 2 and 6 respectlvely.

REVIEI,f OF RESEARCH INTO POKER

1.3 Methods of classlfyine poker-llke sames

Untlt 1928 the mâthematlcal analysls of poken was

llmlted to a probablllstlc and comblnatorlat treatment (see

Borel and VlI1e (3)). In 1928 von Neumann, (26), publlshed.

the flrst paper on game theory and showed how 1t was poss-

lbIe to obtaln exact solutlons to slmple poker-llke games

uslng h1s newly developed theory. Von Neumann formulated

and solved a slmple poker-11ke game, (27), and found that the

solutlon contalned the element of blufflng (see 1.4.1),whlch had prevlously been consldered to lnclude psycholog-

lcal factors, and was therefore consldered not to be a

sultable case for a slmple mathematlcal treatment. (g

further dlscusslon of blufflng ls glven 1n chapter 3).

Slnce then many papers have been publlshed deallng

wlth lncreaslngly sophlstlcated verslons of poker-llke

games, ( t2,13r20 ,2?,24 ,27 ,29) . The recurrlng probrem has

been one of computatlonal d.lfflculty. Thus, although ltcan usually be proved that optlmal strategies to any poker

game ex1st, the mathematlcal technlques necessary to dls-cover them are generally lacklng.

4.

blhen presenting the survey of the llterature on

poker-llke games lt 1s helpful 1f the games are classlfledby the followlng parameters.

,,

(a) Number of players

The number of players largely determlnes the overall

complexlty of the game, as speclal dlfflcultles are

encountered when the number of players exceeds two.

(b) The hand strueture ln the same

The hand structure ln the game 1s usually eltherdlscrete, flnlte and small or else contlnuous, and

plays a large part 1n determlnlng the method of solut-1on. Real poker ls an exceptlon in þhat although the

hands are dlscrete, they are so large ln number, that

the hand structure can be consldened to be contlnuous.

(c) The es nnin bettlnThere is conslderable varlatlon 1n the rules governlng

bettlng. lhe more lnvolved the bettlng, the greater

are the computatlonal d1ff1cultles. Some games have

a great varlety 1n the blddlng and reblddlng, whlle

others are llmlted to one type of bet.(d) The partlcular method used to solve the Rame

There 1s no s1ngIe method whlch 1s applfcable over the

whole range of games. The technlques used to solve

each lndlvlduaI poker-Ilke game are usually dlfferent.

5.

(e) Relatlonshlp to real poker

Games may be further classlfled by the lnslght that

they provlde lnto real poker.

l- 4 Games of hlstorlcal lnterestA number of games that play an lmpontant part ln

past research but that are not dlrectly related to thlg

thesls are now presented.

f,4.1. The games of von Neumann and Morgenstern

Von Neumann, (27), was the flrst wrlter to formulate

and solve a poker-llke game. The rules of thls game allow

two players to obtaln random hands s1 e [0r1] and

s2 € [0r1] where s1 and s2 are unlformly dlstrlbuted

over the closed lnterval [0,1]. Both players bld slmul-

taneously, elther a or b unlts (where a>b and a and

b are flxed) not knowlng the value of the other playerts

b1d. The term 'tb1d slmultaneousl-yrr signifles that the

players make thelr blds together, and each playen has no

knowledge of what h1s opponent w111 b1d. If both b1d the

same amount, hands a?e companed wlth the hlgher hand wlnnlng

the pot. If one blds hlgh, the other 1ow, the player bldd-

lng low has the optlon elther of forfeltlng hls low b1d, oR

of lncreaslng the amount b1d to equal the bld of the other

player, 1n whleh case the hlgher hand wlns ttre pot.

The method of solutlon glven by von Neumann, (27),

(although lt 1s not able to be applled dlrectly to the work

6.

of thls thesls) glves heurlstlc lnslghts lnto how real poker

should be played and shows clearly the advantages of bluff-1ng (see chapter 3).

Blufflng takes place when a player makes a hlgh bld

(a unlts) wlth a hand which 1s not 11kely to wln lf the

other player should look. The blufflng strategy for thlsgame w111 be described preclsely ln chapter 3. Blufffng

has 2 purposes. Flrst, the hlgh bet may force a wlnnlng

sltuatlon by causlng the other player to back down. Seeond,

lf the other player does look, he w111 notlce the bluff.Later, therefore, he 1s more llkely to look at a hlgh bet,

slnce he may feel that 1t 1s another bluff. Thus, bluff-lng w111 tend to ensure that a blgher proflt ls made from

good hands as hlgh bets w111 be more 1lke1y to be looked at.

A modlfled form of thls game aIlows the flrst play-

er to b1d elther a (frfgh¡ or b (fow). If he blds hlgh the

second player has the optlon elther of dropplng out and for-feltlng b units, or of matchlng the flrst playerrs hlgh

þld, wlth the hlgher hand winnlng a unlts. Thls model

ls dlseussed 1n chapter 3.

l.\,2. The Eames of G1111s et a1. and the gane of

Bellman and Blackwell

In thelr paper G1111s, Mayberry and von Neumann,

(13), have solved a two person varlant of a poker Same' wlth

slmultaneous blddlng and a continuous hand structure. It

T.

ls simllar to von Neumannts or1g1naI game, (27), except

that lnstead of allowlng two flxed slzes of a b1d r a and

b, the bid can take any vaJ-ue between a and b. Two

verslons are considered, one where all blds 1n the closed

lnterval [arb] are allowed, and the other $t1th only a

fln1te posslble number of blds 1n the same lntervaI. But

the rules dlffer from von Neumannfs orlglna1 game (1.4.1)

ln that no upgrading of bet by the lower bldder, ls allowed

and the hlgher bldder wins the amount of the low b1d. 'I'f":

blds are equal, the hlgher hand wlns the amount bet.

Bellman and Blackwell , (2) , have al-so solved a

varlant of poker, which has two players, a contlnuous hand

structure, and 1s very slmllar to the galne of von Neumann,

(27) .

The main eharactenlstlcs of the games that have been

descrlbed are:-(1) OnIy 2 partlclpants are allowed (i.e. they are 2-person

games ) .

(2) The hand strueture ls contlnuous.

(3) Bettlng 1s Ilmlted with no reralslng.(4) Although the methods of solutlon are dlfferent, none

ls appllcable to games treated 1n thls study.

(5) The solutlons show the lmportance of blufflng.(6) Ttre solutlons cannot be dlrectly applled to any real

game of poker.

B.

1.4.3. The Eames of Karlln and. RestnepÞc. and PrulttKar}ln and Restreppo, (ZO¡, have developed a flxed

polnt method whlch has appllcatlon to varlous models as

follows.

The first model relates to a 2-person game wlth a con-

tlnuous hand structure and k rounds of bettlng. No other

game has been noted that allows a lange number of raises and

reralses as does thls model. The second model closely

resembles the gane solved by G1111s et.aI. (see 1.4.2).

Prultt , (29¡, has used. the method developed by Kar1ln

and Restreppo, and applied 1t to a contlnuous verslon of stud

poker. Stud poken 1s a varlety of poker that 1s fundament-

ally dlfferent from draw poker , þo whlch thls thesls relates.The maJor point of dlfference between the two games arlses

because certaln cards belonglng to each player are revealed.

ln stud poker, whereas hands are completely concealed lndraw poker.

1.5, Games of direct slEnlflcance to thls thesls

The games that are now presented are of dlrect s1gn1fl-

cance to the development of the work of thls thesls.

1.5.I. Kuhnrs 2-person Eame

Kuhn, (22), has proposed a slmpllfled 2-person poker-

I1ke game whlch lntroduces the concept of behavlour para-

mete::s. Behavlour parameters descrlbe the probablllty wlth

whlch actlons are undertaken 1n any prescrlbed situatlon,

9.

and can be used to slmpl1fy the computatlon lnvolved ln

solvfng a game. (Tfrts concept 1s descrlbed and used 1n

chapter 4).

The game has a dlscrete hand strueture (onty J types

of hands are allowed) and the Ëolutlon exhlblts both bluff-lng and underblddlng. Blufflng 1s a feature whlch was

present 1n the other models descrlbed prevlously, but under-

blddlng has not occurred before. Underblddlng occurs when

a player bets the mlnlmum posslble v¡hlle holdlng a strong

hand. This manoeuvre ls used by experlenced poker players

when holdlng a strong hand, 1n order to entlce the opposlng

pl-ayer lnto naklng a large bet, whlch 1s then ralsed. Ïthas the added advantage that, the opposlng player having

been tnapped ln thls way ls later reluctant to ralse a small

bet, even when he has what ls qulte probably the wlnnlng

hand. '

I.5.2. Nash and Shapleyrs 3-person game

The 3-person game of Nash and ShapIeV, (24¡, 1s lmport-

ant to thls study beeause lt presents the concept of the

equll1br1um polnt (e.p.) Nash, (25), deflned the e.p. 1n

order to solve non-cooperatlve games wlth more than 2 play-

ers, a sltuatfon whlch von Neumann?s orlglnal theory does

not encompass. Nash and Shapleyts game 1s an extenslon of

Kuhnrs 2-person game to the l-pe:rson case although the

number of posslble hands ls reduced from 3 to 2 The

10.

sorutlon obtalned shows exactry the same features as Kuhnfs

game, 1.e. blufflng and underblddlng.

1.6. Fnledmanfs Eame

Frledman, (I2¡, vüas the flrst wrlter to eonslder a

reaIlstlc blufflng sltuatlon, whlch even though very slmpll-f1ed, had a dlrect practlcal applicatlon to certaln sltua-tlons whtch arlse ln neal poker. The g;eneral approach toobtaln these solutlons 1s based on concepts sfunllar to those

used ln thls thesls. The maln features of these games are

as follows.

Frledman flrst conslders the 2-penson game where the

pot contalns k un1ts, and player A holds a 4-flush(f card short of a flush) whlte player B holds 3 of a klnd.Player A discards 1 card and has a chance p s O.Z of making

the frush. Prayer B dlseards 2 cands but h1s chance oflmprovlng 1s small by comparlson (proUabl1lby 0.085). Itls clear to ptayer B that slnce prayer A has bought I card,

then lf he completes hls flush, h€ w111 beat..player Brs hand

unless B lmproves (hfghfy unllkefy). player A 1s flrstto bet, and he can elther decllne to bet on erse bet I unlt(tr¡e maxlmum bet allowed). obvlousry, ln thls sltuatlon,he 1s 1n a strong posltlon to bluff player B (Uy ¡ettlng 1

unlt ) .

Frledman shows that player A should b1uff, when he

mlsses hls f1ush, wlth a probablllty of D

1r+ÐF)-

11.

[obvlously A w111 always bet the maxlmum lf he completes

h1s flushl. Also player B should look at Ars potentlalbluff wlth a pr"obablllty of #. It 1s shown that wlthk=1, one thlrd of Ars bets shouLd be bluffs, and that B

should look at such possible btuffs one half of the tlme.

Frledman next cortslders the same sltuatlon except thatthls tlme player B 1s allowed to reralse I un1t, afterArs bet. Agaln the general concluslon reached ls thatone third of all ralses shourd be bruffs, whlle one half ofall potentlal bluffs should be called. Frledman conJect-

ured that thls generallsed stnategy may also apply to more

compllcated sltuatlons whtch defy analysls.

These solutlons may have a dlrect applleatlon to real.poker, and ar"e slmple to apply ln practlce.

L.7 . Numerlcal technlques

All of the above mentloned games were solved by analyt-ic methods, and none vrere solved numerlcally. By numerlcal

methods 1t ls meant that some sort of lteratlve technlque

1s used whlch can approxlmate the exaet solutlon. The

reason for conslderlng such methods 1s that they may be

lmplemented on the computer and ln thls way allow solutlons

to be obtalned whlch can not be obtalned analyttcally (tfns1s very s1m11ar to the sltuatlon encountered ln the work on

dlfferentlar equatlons, where lteratlve numerlcar methods

play an important rote).

L2.

One of the maJor tasks of thls thesls was to flnde.p.rs for certaln n-person games. However, 1n the course

of thls work 1t was found that the equatlons obtalned ürere

too complleated to be solved analytlca1ly.Only I numerf-c method sultable for use wlth thls

problem was dlseovened ln the llterature. Thls was the

method of Rosen, (31), whlch ltenated to flnd the solutlonpolnt by ealculatlng eertaln derlvatlves. Ttrl-s

method ls dlscussed (cf¡apter 4) and shown to be unsultable

fon the games consldered hene.

A survey of the llterature showed that very I1tt1ework has been carrled out 1n the area of numerlcar methods.

However, the usefulness of thls approach w111 be demonstrat-

ed later ln thls study (chapter 5) when results are obtaln-ed whlch could not be obtalned analytlcally.1.8. Computer slmu1atlon of poker

slmulatlon mlght be expected to provld.e an effectlveapproach to the solutlon of poker-rlke rgane.s. However, theonly worken whose publlcatlons develop.thls approach 1s

Flndrer. Flndrer, (ro), gave a flow-chart for a proposed

poker playlng program, and recently pubrlshed, papers (tt,;r,'is)whleh lndlcate that further work 1s currently belng carrledout 1n thls area. some attentlon has been glven 1n thlsthesls to the slmulatlon of poker as noted 1n the folrowlngsectlon.

13.

NEhI WORK PRESENTED IN THÏS IS

1.9 Poken slmulatlon

As a part of thls study a poker simr¡lator was

programmed. The poker slmutatlon 1s deseribed 1n chapter

2 and 1s related to the maln body of the thesls 1n the

followlng ways.

Flrst, even though lnltlally lt was hoped that

slmulatlon could be used to asslst ln the analysls of

poker, 1t was estab]1shed that thls was not a sultableparlrcr^Iar

nethod fon the purposes of thls\work. Secondly, certaln

analytlc results which are obtalned later may be employed

1n thls program to enable !t to play a better game of poken.

Thlrdly, the use of lnteractlve programs ln the analysls of

poker are dlscussed. Furthermore, as a byproduct of thls

work, certaln poker probabllltles, whlch mlght be of

lnterest to other workers ln thls f1eld, and to the practic-

al poker pLayer, were evaluated.

1.10. A new solutlon method for n-person games

As has been mentloned 1n sectlon L.7 ' a sultable

numericaL method for solvlng the class of ga¡nes consldered

here, could not be found 1n the llterature. As a part of

thls study a new algorlthm vtas formulated. to meet thls ,r"".i,

and. lt has been used extenslvely throughout thls thesls.

14.

1.11. Evaluatlon of a S Declal lnteEralDurlng the evaluatlon of the payoff functions for

the poker-llke games consldered here, a partlcurar multl-dlmenslonal lntegral was found to oceuf repeatedly. Thls

lntegral, ls of central lmportance to thls thesls and as 1t

has not been evaluated ersewhere lt ls evaluated here (see

Appendix A).

L.12. Formulatlon and solutlon of a new poker-llke trame

In ehapter 5 a 4-person poker-Ilke Bame ls formulat-ed and solved. It ls consldered that thls game ls a s1g-

nlflcant contrlbutlon to the results already achleved lnthls area of study, for the followlng reasons

(a) It 1s the flrst poker-Ilke game solved whlch a1lows

more than 3 players (a 4-person game 1s consldered).(b) It has the unlque property that 1t - con hø related

to a large subset of a commonly played varlety ofpoker. It ls noteworthy that the solutlons found

agree closely wlth strategles eommonly used by

experlenced players.

1.13. Appllcatlons

An appllcatlon of game theoretlc methods to a network

1s presented 1n whlch a new crlterlon of optlmallty 1s

deflned. Thls allows the network to be optlrrlzed uslng

methods developed ln thts study.

15.

Even though the slmllarlty of poker to business slüu-

atlons has been noted by Pruitt, (29), and Kar11n, (19), no

expllcl-t examples eould be found ln the llterpture. Thus,

as a sequel to thls study, a buslness operatlon, whlch ls

dlrectly related tö poker, 1s deflned ancl solved,

In ttils way 1t ls shown that the maln work of thlsthesls may be applled to other, not dlrectly related

pnoblems.

LAYOUT OF THESIS

1.14. Layout of thesls

Thls thesls has been organlzed 1n the folLowlng way.

Chapten I deflnes the pnoblem consldered, presents a

lltenature survey, and descrlbes new work carrled out.

Chapter 2 describes the computer slmulatlon of a commonly

played varlety of poker, and lts relevance to thls work.

ïn chapter 3 a mathematlcal statement of the problem of

solvlng n-person games 1s glven and then a new algorlthm,

speclflcally deslgned to solve the games treated ln thlsthesLs, ls presented.

In chapters 4 and 5, 2 13 and 4-person verslons of the

game slmulated (but unsolved) 1n chapter 2 are consldered,

and solutlons found.'

Chapter 6 presents examples of practical appllcatlons

of the work carrled out here. Chapter J summarlzes the

work done, draws concluslons, and lndlcates posslble new

16.

areas of research and app11catlon.

The subJects assoclated wlth the evaluatlon of the

lntegrals are treated ln chapters 3r4 and 5 whlle the

evaluatlon of the lntegral 1s presented ln appendlx A.

]-7.

CHAPTER 2

SIMULATION OF POKER

INTRODUCTION

2,L. Introductlon

Some work haS been done ln thls study on the slmula-

tlon of poker but, for reasons dlscussed 1n detall l-ater ln

thls chapter, the research was not completed. However,

durlng the research, results nere obtalned as follows:-( 1) pnobabltltles were deflned and calculated of

wlnnlng wlth speclflc hands, both before and after

the draw, wlth glven numbers of players

(2) values were obtalned fon eomputatlon tlmes for

lnvestlgatlng dlfferent poker strategles by slmulatlon

(3) the strengths and weaknesses of lnvestlgating poker

by lnteractlve slmulatlon were assessed by means of a

pllot study.

These results are considered to be of sufflclent

value to Justlfy a dlscussÍon of the work done on the slmu-

Iat1on. Thls dlscusslon 1s lntnoduced by presentlng

brlefly the prlnclples used 1n slmulatlng a game of poker.

2.2. Prlnclples of poker slmulatlon

The rules of the game slmulated are glven ln some

detall ln sectlon 2.5, but for general explanatory purposes

a brlef descrlptlon of the game slmulated ls as follows.

Flve cards are dealt to each of up to 7 players. Players

18.

conslder the varue of thelr hands (aef,lned by comblnatlons

of paÍrs, three of a klnd eto. as descrlbed 1ater) and may,

by paylng money to enter, take part 1n the game. Havlng

entered, a player has the optlon of neplaclng up to 4 cards

ft"om hls hand 1n onder to lmprove h1s hand. hlhen arr play:ers have .exerclsed thelr optlon a glven player (tfre player

to the left of the dealer) ûâV¡ 1f he wishes, bet an amount

on h1s hand or drop out. If he bets the next player may

nalse the bet, ot, by bettlng an equal amount stake a clalm

to |tseerr hts predecessorts hand, op drop out. llhe opportun-

1ty to exerclse these options passes from player to player

untll all players but one have dropped out or, untl1 allplayers but one, have claimed to see the hand of the remaln-

lng player. In the latter lnstance the player seen must

show h1s hand and the monles staked go to hlm unless some

other player lays down a stronger hand" In the former

lnstance the resldual prayer takes the pot wlthout showlng

hls hand. Because a player may w1n wlthout showlng hlshand there 1s the opportunlty for a player to wln by bluff-lng, that ls by g1v1ng an lmpresslon of strength sufftclentto frlghten h1s opponents out of the game.

The slmulatlon of the game may be undertaken lnphases thus:

(1) Generatlon of hands

The flrst step 1n slmulatlng a game of poker 1s toagree on a representatlon of each card by an lnteger from

L9,1 to 52, thus:

Ace of Clubs

Two of Clubs

=1-I

=2

Queen of Spades = 5I

King of Spades = 5Z

These 52 diglts are stored 1n an array and a shuffllngalgorlthm applled, whlch uses a random number generator,

and whlch guarantees that every posslble comblnatlon of

52 cards ls equally llkely. Thls algorlthm ls descrlbed

ln sectlon 2.6, This representatlon of a shuffled deck

of cards may now be used to d.ea1 each player a hand of 5

cards, and to deal ::eplacement cards.

(2) Enterlng the game

In order to declde whether to enter, the probabll-

lty of wlnnlng before the draw must be calculated. I^llth

thls knowledge 1t ls posslble to deflne probablllty values

whlch will be criterla for enterlng or dropping out. Thus,

1f at a glven stage of the game a hand that has the option

of enterlng has a probablllty of 0.80 (say) or more assocl-

ated wlth 1t, then lt could be asserted that that hand

would contlnue. The probablllties for each hand would

then be comparedr ln a s1mlIar way wlth these crlterla to

declde the fate of that partlcular hand at tlme of pIay.

20.

(3) Improvement of hand

It 1s easy to define a determ1.nlstlc algorlthm by

whlch, glven any hand, the course to follow w111 be specl-

fled. For example, 1f a hand contalns 2 palrs, then the

rules mlghb prescrlbe the dlscardlng of the non-palred card.

A determinlstlc algonithm of thfs type 1s descrlbed laterln thls chapter. Obvlous1y the element of blufflng could

be lntrod.uced lnto thls stage of the gane by speclfylng,

aecordlng to the value of a random number generated, that

a dlfferent algorlthm mlght, or mlght not be used. Thls

second algorlthm might, for example, be designed to glve

bhe lmpresslon that a poor hand was good.

( 4) aetttne

The stage of the game concerned wlth the bettlngwould be slmllar to (2) above. In the same way as ln (Z)

1t would be needed to know a probabltlty, 1n thls lnstanee

the probablllty of wlnnlng after lmprovement. Aga1n, âs

ln (2) a probablllstlc system for bettlng would be requlred..

In thls lnstance, however, more sophlstlcated bettlng ruleswould be needed, based on the probabllltles of wlnnlng afterhand lmprovement. For example lt would be needed to know

how fan bettlng should be taken on a glven hand - 1.e. how

many rounds of bettlng should be entered 1nto.

(5) Determlnatfon of the wlnner

Provlded that more than one hand is left 1n the game

2L.

at completlon of bettlng, the method of wlnner Ídentif1-ca-

tlon ls slnple. Slnce each hand ls known speclflcally, 1t

ls merely a matter of comparlng hands and selectLng the

strongest.

2.3 Results achleved by other workers

A search of the llterature revealed that Flndler,

(:i r ,lgrlOrlt).ris.the only wrlter who has considered the slmulatlon

of poker. Flndlerrs flrst paper, (10), presented a flow

dlagram of a proposed poker playlng program. The program

presented here 1s, ln broad detall, slmlIar to the program

suggested by Flndler.Ii:rdlerrs later work uses the si:nultrtion of polcer as a means of

studying d.ecislon matcing. A detailed cx¡¡;rination of thls work i-s not

includ.ed here for tlvo reasons. First, these paperÊ v'¿ere not available

at the time of vrr.iting this study, arid secondly, Findlerts resufts do

not affect the vaLidity of the d.eductlons made he¡e.

2,4 Factors lnvolved 1n settlnq uD a simulatlon

The maln actlvltles lnvolved 1n settlng up a polrer

slmulatlon are the fo}lowlng.(1) Represent each of the 5Z cards 1n the deek

by a unlque lnteger from I to 52.

(2) Develop algorlthms whlch w111 shuffle thls

deck, and calculate varlous probabilltles

assoclated wlth a hand of cards thab are

cruclal to the game of poker.

22.

(3) Prepare an algorlthm whieh w111 calculate whlch

cands to dlscand durlng the hand lmprovement

process.

(4) Now ñethods of descrlblng bettlng strategles

must be found whlch are lntelllglble to a

computer. The method used here 1s based on

the program belng abLe to evaluate pnobablllt-

les relatlng to a poker hand, then employing

algorlthms whlch calculate when and how much

to bet 1n a glven sltuatlon for a glven strength

of hand. These algorlthms glve numerlcal

procedunes whlch calculate the slze of the beü,

uslng the above mentloned crlterla, and are

based on observed patterns of play followed

by experlenced poker players, (30).

(5) Uslng (1) to (4) above lt ls a slmple matter

to construct a poker playlng program, whlch

must then be ver1f1ed.

DETAILED trMTHOD OF SIMULATION

2.5 Rules of the game

Poker l-s a game played wlth a deck of 52 cards, and

any comblnation of 5 cards eonstltutes a hand. There ls a

¡l,erarchy of hands whlch ls well known and ls glven ln table

1. The partlcular game slmulated here ls called 5 card

draw poker and is the most commonly played verslon of thegame. ft has the fo1low1ng rules.

23.

TABLE ICTASSIF'YING POKER HANDS

Poker hands may be classlfled lnto 9 dlfferenttypes as glven below ln deereaslng order of strength.

It should be noted that apart from the flush and.the

routlne, sults are lrrelevant to hand strength.

1. Stralght flush or routlne, whleh contalns 5 cards insequence, and 1n the same su1t.

2. Four of a klnd, and I odd card.

3. FulI house whlch has 3 of one klnd, and 2 of anoüher.

4. Flush, whlch has 5 cards of I su1t.

5. Stralght, whlch has any 5 cards 1n sequence.

6. Three of a k1nd, and 2 1dIe cards. (IdIe cards are oneswhlch take no part In determlnlng hand strength. )

7. Two pairs and I ldle card.

B. One palr and 3 ldIe cards

9. No palr and 5 ldle cards.

The rules for determinlng the stnonger of 2 hands

when both are of the same type are well known and may be

found ln any book on poker (see (30)).

2\.

There are n players (f s n <

consecutlvely by the lntegers I to n 1n a cloekwlse

manner (Iooklng on to the table from above). P1ayer

n-I ls called the dealer and player n the bllnd.

The dealef shuffles the deck of cands and deals 5

cards to each player. The bllnd now places a compulsory

bet of I unlt lnto the pot. Next a1I players look at thelrhands, and decide 1n turn, beglnnlng wlth player 1 and end-

1ng wlth the dealer, whether or not to play. If they wlsh

to play they must place 2 unlts ln the pot. If no player

decldes to play, the bllnd netrleves h1s I unlt bet and the

game ends, otherwlse the bllnd has 3 optlons open to him.

(1) He may drop out of the game and forfelt hÍs I unlt bet.

(2) He may elect to play on by maklng an addltlonal bet ofI unlt.

(3) He may bet a further 3 unlts (thus maklng a total of 4

unlts ) .

Thls ls called doubllng, and ln thls case each player,

1n ascendlng numenlcal orden, must elther drop out and for-felt h1s bet on else place a further bet of 2 unlts lnto the

pot. ïf, after a double, all other players dnop out, then

25.

the game ends and the bllnd wlns the pot.

Next, each player 1n turnr mêÍ discard up to 4 cards

from h1s deck and recelve an equlvalent numbêr of cards from

the dealer. tfris wlII be referred to as the hand lmpr"ove-

ment stage and completes the flrst phase of the game.

In the second phase of the game aJ-I remaÍning players

take part 1n a round of betting that determines the ultlmate

wlnner. Betting beglns with player 1 and proceeds ln a

cloc},n¡rlse manner wlth each player 1n turn havlng 3 optlons.

(1) He may drop out of the hand.

(2) He may rrlookrr, that ls make such a bet that h1s

total bet becomes equal to the greatest total bet

made by any other playen stl1l 1n the game.

(3) He may rrralserr, that ls bet sufflclent to 1ook,

and then lncrease thls by any amount from I to 5

un1ts.

Untll some player makes an 1nltlal- bet of 1 to 5 units, allplayers whottpasstt(1.e. do not bet) must drop out of the

game.

Bettlng 1s contlnued unt1l either only L player remalns

1n the game, 1n whlch case he is declared the wlnner, or else

untll, after a ralse by one of the players, there has been

no further ralse by the tlme bettlng returns to that player.

fn the second case the wlnner 1s the player showlng the best

hand.

z6'In the succeed.lng game the b1lnd becomes the new

dealer and the other players are approprlately renumbered.

2.6 Slrnulatlon of shuf flln e

Each card ls represented by a unlque lnteger from Ito 52, Inltlally, these lntegers may be stored 1n any

order 1n an array, ICARD(I)....ICARD(52¡, An algorithm

was wrltten whlch used a random nunlber generator to shufflethe d1g1ts randomly in the array ICARD. fhe algorlthm was

based on a method proposed by de Balblne, (1) and has the

property that after the shuffllng, all posslble comblnatlons

of the 52 cards are equarty 11kely. A flow dlagram of thlsalgorlthm ls glven 1n flgure I 1n the form apprled 1n thlsstudy.

This algorithm works by exchanging the first element

in the array IC.ARD( 1 ) , . . .IC¿RD( 52) with it self or withany other element on the right, with equal probability forall exchanges (tfre particular element exchanged. is d.etenmlned.

by the rand.om number: x). This process 1s repeated. wil:1, tTte

second, thind., fourth element etc" up to the second. 1ast,

element. de Balblne shows that¡ regärdless of the lnltlalord.ering, this ¡nethod. w111 yield. a rand.om permutation of

the original arcry, with all possible permutations being

equally likely. Once the d.eck is shuffled. the cards are

d.ea1t ûo ttæ players in the conventional way, one card. per

player, until each player hold.s 5 cards.

lç <- icard-1icard.l <- icard.nicard.n : k

n <- x(52-r) + r

ê random numbe

wlth 0 (x (1

r=r, , , ,52

lcardlei

=L r, , .52

START

FIGUBE 1 27

DT.]CK SHTIF'FT,TNG ATÆORTT]NI

- Set up the integers

I-52 In the arraYicard1,...1card52.Thls represents the unshuffled deck

Conslder the 1th card.

x 1s set to a random unlformlYdlstrlbuted number 1n the closedlnterval [0 ,1] .

x 1s used to calculate a random

i-nteger n where, because of themethod of calculatlon, n lies 1n

the range from i to !2, wlth equalprobabllity for a1f numbers.

For example, i=8, x=0 . 5 gives n=30

Now the ith card 1s lnterchanged wlththe nth card.Thls forms the basis of Balbinersalgorithm, as he shows that at thecompletlon of this Procedure allpossible orderÌngs of the deck are

equaffy l1ke1y.

At the completlon of the algorlthm thedeck stored 1n lcard1 r. . .lcardsz hasheen r.¡ndomlv shuffled.

STOP

28,

2.7 Imprpvement of hand.

Arr important matte:r ln a hand. of Boker is choosing

whlch card.s to replace in the tand. lmprovement qtage.

There is general agreement amongst card players eoncernlng

which card.s to d.lscard. fron any given hand., and. an account

of thls is given i-n Reece and. TVatklns, (:O¡.

thus it 1s commonly accepted. that' if a hand. cont-.

ains 2 palred. card.s, and. J non-paired. or id.Ie card.s, then

it 1s best to d.iscard. the 3 id.le cand.s. A¡r aggregation of

these accepted. norms would. make it posslble to construct a

d.eterninistlc algorithn to compute which card.s should. be

Oisearded. for any given hand.. Howeven, before using

these arbitrary Jud.gements, a Monte-Gar1o method. was

Brogrammed. to d.etermlne optimal throwaway strategles forêvery type of hand-. The roethod. used. to achÍeve thi-s was

based. on an id.ea of Flnd.lerts, (tO). The best throwaway

comblnation for a particular hand. was d.etermined. by consid.-

ering all posslble throwaway combinations, impnoving tJre

hand. rand.omly a lange number of tlmes (:2OrOOO) for each

of these, and. then choosing that thnowaway which gave the

best result. The nesults obtained. agreed. exactly wlth the

throwaway strategies ad.vocated þy ercperienced. players (:O¡.

However, the time taken to compute each case u.sing tltismethod. was approximately 2O second.s of central processor

(".p.) tlnef , and. hence a d.etermlnlstic algorithrn was trled..

All computation was calrried. out using a CDC 6¿+00 computer,and tlmee talren aBply to this machine (see (4)).

29.

this new algorlthm worked. in the foJ-lowing way.

First a hand. was cl-assified accord.ing to its type, and. then

the appropriate card"s to d.iscard. were chosen on the basis

of the results that had. been obtained using the Monte-Carlo

method.. This algorithm was faster by 2 ord.ers of magnlt-

ud.e than the Monto-Can1o method., and. it is d.escribed. in

more d.etail in the flow d.iagram given in figure 2.

2.8 Calculation of probabil-ities

The calculation of the probability of winnlng with a

certain hand., both before and aften the d.raw, 1s crucial inpoker. Find.ler, ( t O) , suggests a Monte-Carlo method .

will be qiven 1aten) " but found- to,e,tt""-¿jiu( raf[,-"1t, t¡ "]d.do¡s J. 1).t"< .v ¿,(d *+ b< o ü: i'a^¿u <9

1ve1y, Epstein, (g), d.erived the

table of probabilities given in table 2. However these

are not sufficient ln themselves, because although the

table gives the probability of obtaining any glven hand.

and. the probabillty of lnproving to any better hand. forany giver nu¡rber of card.s clrawn (aiscard.ed.), Ít d.oes not

give the corresponiLlng probabllities of winning. As a

table giving the required. probabillties could. not be

found., they rÀrere eval-uated in the followillg tJtrâ]ro

FIGURE 2

ALGORITHM TO DETERMINE WHICH CARDS

TO DISCARD DURING HAND IMPROVEMENT

30

STOP

must bel+ of a klnd ¡no card.e dis.cudeil

no cerd.Btli s cêrd etl

ful-1house

discerd. the 2cards not in-clud.etl ln 3a kinat

3ofa klnd

lqrest lrcerdls tÉa-cütleil

discerd 1 non.paired card2 paþ6

11¡ l+ fruhstralghttlls ce¡ded

discartt the 3non-paired

cardsor straightflush

no cartlatlisce¡cled

straightor flush flwh

straightstralght flushto ne,]{ediscerd I card

eS¡ pei¡edcætl,s

dlscard 1card to mslteflush

1 pair &pair<9L-fluBh

Stüt

Orlgiqalhar¡(L

One paÍr

trro pafrs

3ofakintl.

straight 0.0039

flt:sh 0.0020

full-house 0.00Ih

h of a kind O.OOO2L

strat@t 0. OOOOI¡+fh¡shroyal flush 0.

h straisþt 0.035( open)

31.

Probability of,reeeivinF inprovetl

hantl

0.160.f,j.l+0.01020.0028o.28'lo.LTz0.0780.00830.0009o.26

o.oB5

0.0610.0h30.101+0.06h0.0210.085

fuIL house

cannot be Ímprovecl

It

ll It ll

straÍgþt

straigþt

flush

stralgþt fh:sh

an¡r straight orflush

TABIE 2

Probability Ca,qcls Inprovett ha¡¡ctof recelv- tl¡avnins that ban¿|.

0,1+226 3 tryo pairs3 3of akfncl3 full hor¡ee3 hof akincl3 any J.rprorrement2 2 pairs2 3of aklntt2 fult hor¡se2 4ofakind

,, 2 an¡r inprovement

0.0l+?5

0.02Ir firl1 housel+ of a kinttany improvementfr¡LI house\ of a kinctany lmprovement

l+ straÍght( e"p)h flush

h straigþtfl-ush

0,123

0.01+3

0.000123

0.0039

0.0020

0.001h

0.0021+

0.00001¡+

. 0.00000123

o.17

0.085

0.191

0.01+3

0.319

I

222III

0

0

0

0

0

0

1

I

L

I

1

ll

il

It

ll

ll

tt

lt

il

Itil

ì

32.

Let h be some unlmproved poker hand, then f¡(fr)

is the probablllty that the hand h has of beatlng any

other random unlmproved hand. Now lf h ls some already

lmproved poker hand then define f"(n) as the probablllty

that the hand h has of beatlng any other random improved

hand. A qui-ck method of evaluatlng f*(r,) and f¡(h) f s

requlred.

F1ndIer, (10), proposed that a Monte-Carlo method

be used. Thls method was programmed but found to be slow

(with each calculatlon taklng about 5 secs. of c.p. tlme).f NsEr &ofç Feú-r=.

Findler 1n a later publication, (ll), deals with the calc-

ulation of probabilities i-n a mor'e sophisticated way. However,

since this vrork was not a.vailable at the time of lvritinE; the sim-

ulatíon described here, use of these new method.s could not be tnad.e.

This does not affect the results of thís work as the exact methocì

of calculating probabllities is not of central importance. The

approach to the calculat'lon of probabiJ-ities. adopted. in thÍs study

is a tv,lo-phase one, and. will now be d.escribed.- ---,,, t v¡¡ q¡¡J l¡qllu ¡. t - u \--. , ----ù

approximately determlned thus. First, a large number,

N(N=301000), of random hands are generated. Second, L,

the number of tlmes that the glven hand h beats the random-

ly generated hand.s ls noted, and fu(f¡) ls approxlmated.

by t/N. Even though the lmplementatlon of thls method 1s

stralghtforward, the calculation must be carrled out ln such

a way that the executlon tlme fs kept to a reasonable leveI.

Because of thls constralnt varlous compllcatlons ar1se, and

33.

the teehnieal. detalls of thls process are presented 1n

appendlx C.

The Monte-CarLo method used to evaluate fa(h)

dlffers from the method glven above only ln that the hand

h 1s tested agalnst a lange numben of randomly lmproved

hands (see appendlx C).

The lnterpolatlve phase of thls process w111 now be

des cribed .

Table 3 deflnes 9 lntervals 1n whlch any unlmproved

hand h may 11e. For each of these lntervals the table

glves hi whlch ls the lowest hand of the i-th 1ntervaI,

and hî which is the hlghest hand of this lnterval (lncIus-

lve). The correspondlng probablIltles fr(nì) and r¡(rrÎ),

as calculated by the Monte-Carlo method are also given.

Conslder, for example, the second 1nterva1. This

interval lncludes all hands that contaln exactly 1 pair.

By consldering all hands of thls type it can be seen tTrat

the lowest posslble hand. of thls type 1s 2C 2D 3C 4C 5C

whlLe the hlghest hand ls AC AS KC QC 'rC.

The probablllty of wlnnlng that any unlmproved hand

h has may be approxlmated from thls table |n the followlng

way. First lt 1s requlred to determlne the lnterval 1

to whlch thls hand belongs and correspondlng to thls lnter-

val there ls a numberlng functlon pi (see table 4) that

asslgns an lnteger pi(h) to each hand h belonglng to

the lnterval. Thls functlon pi ls chosen 1n such a

Inter-va1 No.

34.

TABLE 3

PROBA3ILITIES OF !ÍTNNING FOR ITNIMPBO\TED

HANDS CATCULATED BY A MONTÈ.CARIO [4EfHOD

bl:Low hantL of inter-vaI

rr (rr| ¡ hl:higb hana of l:1."r-

ro(nl)

o. h9\5

0.9198

0.97Ir

o.99zo

0.99\\

0.9975

o.9993

1.0000

1.0000

Note that even thougþ suÍts ôo not influence the strength of a poker

bancl, they are incluctecl in the above table for greater clarity.

1

2

3

l+

5

6

7

I9

7C

2C

3C

2C

5C

7C

2C

2C

5c

l+s

2D

3S

2S

)+o

6c

2S

2S

l+c

3D

3C

2C

2D

3S

5C

2D

2\T

3C

2D 5S

l+C ,C

2S \C¡+c 55

2C .A,S

)+C 2C

3S 3C

2D 3S

2C AC

QC

KC

KC

AD

QS

JC

AD

AD

8C

JC

qc

I(S

2C

JC

10c

KC

ÆT

JC

9o

JC

2C

3S

L0c

9c

KS

KS

10c

AC KC

AC AS

AC A,S

AC AS

AC KD

AC KC

AC AS

AC ÀS

AC KC

0.0000

1.0000

0.9993

o.9982

o.996t+

0.99¡+o

0.9?1r

o.92O'

0.5010

35..

TA3I,E \qAtvo uuærnruo rutcrrqt pr(rr)

pi(tr) for h in group Í

pi(h) = fr

pz(h) = it

ps(rr) = Gk?@ *t_,

B,.(h) = it

ps(h) = il

pe(h) = il

pz(tr) = L2(íL-z)+iz

pa(h) = it

il=numerical valueof hÍghest card

il=val-ue of pair

iI=value of higher pafri2=value of loner pair

iI=value of 3 of a kinct

iI=value of higþ carcl(the one exceptionalcase when Ace is eor¡nt-ecl as l- rather tha¡r 14oceurs when the Aceforns the first carcl ofa straight

il=value of hÍgh earcl

i1=va1ue of 3 of a kincli2=value of 2 of kind.

as for straigtrt withil=va-lue of hÍgh carcl

nothing

I pair

2 pair

3 of a kincl.

straigþt

flush

fi¡ll-book

routine

L

2

3

l+

5

6

T

I

variable tlescriptionHantl tyte

36.way that the value of ft(fr) may be approxlmated by a 11n-

ear lnterpolatlon between fr(rri) and fr(ni), see dlagram

below, where the gradlents AB and BC may dlffer sllghtly.

ru(rrl)

fu(rr)

fb (hi )

nr(nl) pr(rr) pr(r'íl

By assumlng that the gnadlents AB and BC are equal 1t

may be found that

c

B

A

f¡ (h) = fb(ht) +pr (h) - pr (r't)

p,(r.?) - p,(ni) ' [ro(r'?) - roCni)l

where fu (h) approxinates the required. probability.For example, in the lnterval

i=2, where hi = 2C 2D 3C ¿p DC

and. h? = AC ^eS KC QC ,TO

it 1s natural to choose pr (h) equal to the numerlcal

value of the pair, where ttre ace is counted. as Ilf and. itfollows that pr(ht) = 2 and pr(h?) = 14.

Hence, given some hand. h in this lntervalr sâJIr

h=6c6o7c2DKg

J'.

fo (h) is appro mated. by

pr (: ) - pr (rr1)r¡(rrt) * -p, (rt?)- pr (ttt )

#. [0. g19l-o.5oto]

. lfb (b?)-f (hi) ]

= 0.5010 +

= o.6z6j.

Note that the fail-ure of thls approximation to d.istinguish,

for example, b etween 6c 6o 7c 2D KS and. 6¡t 6s ¿c QS ffi'

is unimportant because the J non-paired. card.s in each hand'

are later to be d.1scarded., and. thus may be safely ignoned..

Winsrlng probabili ty for an alread-v improved. hand., f"(h).

The wlruring probabillty for an alreaÖy improved-

hand. f.(ft), is found. by the same lnterpolatlve rnethod. as

was used. to flnd- fo (h) except, iî this case table 5

d.efines the lntervals used..

2.9 BettinE strategies

the betting may be il-ivid.ed. into 2 phases, betting

before the d.raw, and. bettlng after the d-raw.

In betting before the draw each player d-ecides, in

tr:rn, whether to ante 2 units and p1ay, oP whether to d'rop

out of the hand.. This d.ecision is d.etermlned- by the cards

that a player hold-s, the number of players who have alread-y

anted., the number of players still to d.ecld-e, and the

state of the ganer

TVhen this progrsln was first written, a very simple

38.

TABI,E 'pnoBABILrúrns or !ÍrNNrNG FoR rMpnovgD HAI{DS

c.ql.,culAtgp gy .A l,{oNTE-cAdLo t'æTHoo

Inter-val no. hl : ro,r har¡rl of *ïí- rrf :rrigh Hand of t"l;i-fè(hi) få(hï)

t2

3

l+

5

6

7

I9

0.50r.1

o.7983

0.9097

o.97L6

0.9820

o.98l+6

o.9967

o.9997

1.0000

AC KC 8C JC 9O

AC AS KS QC JC

AC AS KC KS 2C

AC AS AD 2C 3S

AC KD 8S JC r0S

AC KC QC JC 9C

AC AS AD KC K5

AC AS AH AD KS

AC KC Qe JC 10C

0.0000

o.9999

o.9967

o.gB52

0.9820

o.9778

0.9100

o.7997

o.5222

7c

2C

3C

2C

5c

7c

2C

2C

5C

l+s

2S

3S

2S

4o

6c

¿Ð

2S

l+c

3D

3C

2C

2D

3S

5C

2D

2H

3C

2D

hs

2S

l+c

2e

¡+c

3S

2D

2e

5B

5C

l+c

5c

AS

2C

3C

3S

AC

39.

crj"terion for entering or d.roBping out was used.. However,

this Ìuas d.iscard.ed., because Later work in this thesis (see

5.5) shovuecL that there was a better approach to this prob-

Iem, and. since littl-e extra work was involved., the program

was fnodified. to operate on tlrese improved- principles.

the strategy, a,s used in the program, 1s most convenj-ently

consid.ered. in two separate parts.

(t) The minimum hand- required. to enter the ga{ne when no

other player has yet en@.

Suppose that a player hold.s hand. h, no other

player has yet entered. the garne, anÖ remalning players have

yet to d.ecid.e. Then, (u" shown in chapter 5) it is best

to enter the gane only if f¡ (ft) >

(z)

f (n) = o.o1n2 + o" 13n + o.5 (t)urfur¿ n¡ 'ts- l-l^. u.-*^\-" o{ p\a1-."3.

Minimum hand. reoulred- to enter if at least one other

player_Àas alread.y entered.

The minimum hand. nequired to enter lf at least one

other player has alread.y entered. is calcul-ated. in the

following ïray. A player should. only enter if his hand. h

is such that it is significantly better than the minlmum

hand., H, that the last player to enter is expected to have.

More specificallyr âs shown in chapt er 5, a hand. h is

good- enough if f¡ (n) >

a=f¡(U)

40.

an.l s(N) = ¿*I(t-¿). (z)5'The pnactical applicatlon of this idea is llIustrated. by

means of a simple eXample.

Consld.er a 7-player game where the players ho1d.

hand.s where probabili.ties of winnlng, fu (h), are as

given below.

Player number Nunber left to declare f"(þl1 6 0.97

2 5 0,gg

3 4 0.87

4 3 0.96

5 2 O.52

6 1 0.76

7 o 0.84

The d.ecisions mad.e by each player are summarised. and

explained. 1n table 6.

Playler n (n=7 in the example above), is called.

the lfblind'|, and. accord.ing to the rules of the game has 5options avail"able to him, to d.rop out, play on, or d.ouble.

He chooses to select his strategy in the folLowlng way (see

chapter þ).

Drop

PIay on

if

if

ifPouble

4r.

r¡gr,E 6

AI{ E)OMPLE OF TTIE BETTING AT,GORITTIM

T player ?, the t'blÍndtt, selects his strategr ln a different way,

which is tleseribetl on the previous page.

l, becomee O.92

0 beeomes 0.96

I remains at 0.96

L remains at 0.96

f, renains at 0.96

I renalns at 0.96

becar¡se

becauge

beear¡se

becauge

beear.¡se

beeause

0.97>f(6) =o.92

0.99>e( c)=e( o,92)

=O.96

0.8?<e(s)=e(0.g6)

=0.97

0.96<e(ø)=s(0.96)

=O.97

o.52<e(r )=e( 0.96)

=O,97

o.?6<s(c)=s(0.96)

=0.9?

enters

enters

d.rops

drops

d.rops

d.rops

I

2

3

l+

5

6

Í,, tbe orobability ofthe mÍninr¡n requirerlhar¡tl. heltl by last plav-er to enter

ReasonDecisÍonmade

42"

It 1s aJ-so shown in chapter 5 that following a

d.ouble it is best for the players still 1n the game to

continue playing and this strategy is used. hene.

Second. round. of bettlnE

Now the second. round of betting will þe considered..

As this part of the game has not been solved. ana1ytically

d.ue to its complexlty, the following strategy, partiallybased. on ldeas proposed. by Find-ler, (tO), and. Coff in, (6),

is used. 1n this program.

I,et på(h) be the probabillty that a Ì:and. h lrae

of winning, after the d.raw, agalnst n other players,

where pË(h) = [f"(h)]n. Suppose that the size of the

pot 1s p¡ q. is the amount required- to 1ook, and. e=1

1s some eonstant. Then the strategy followed. is:a player dnops if pâ (h) . (p*q.) <

a player looks if q-e < pâ (ft) . (p*q.) < q+e

a pì,ayer ralses by

på(h). (p+q.) q units if çr+e < på (¡r). (p+q.) 3)The strategy 1s based. ona computation of a playerrs

expectation for a given hand. (expectation equals probabil-

ity of winning x total- size of pot), which 1s then compared

with gr the amount required. to Look. As this e:cpecta'

tlon is below, approximately equal to, or above er so a

player d.rops, looks or raises. Obvior:s1y the value of e

can be ad.justed. so that a player will raise more or less

frequently.

4¡.lfrls very slmpJ-e algorltJrm d.oes not includ.e bluffing

althougþ this was includ.ed ln an earller version of the

program, but removed. because the program was too slow

(see next sectlon).

Proeramming

The implementatlon (programmfng) of the simulation

was a simple though ted.ious rnatter, and hence f\¡¡thercleta1ls will not be glven here, ft was e]cperimentally

found that each game simulation took approximately fr ""..of central processor (".p.) tlme.

DTSCUSSTON

2.10 Checking of results fro4_sellpuiLgr runs

Before the results obtalned. ïuere used., a check ïuas

carrÍed. out to confirm that the program was fl¡nctioningcorrectly. This Tiras done by simulatlng a nr¡mber of games

(about 60) and. printing out al-l the relevarrt d-etaiLs of each

game. ït was found. that the plays mad.e in each game Ìuere

consistent with the loglc r:sed. in the simulation.

2.11 Interactive poker

The program vyas set up to play poker interactively,with a human player able to take the part of one or more

of the simulated players. Communicatlon between the

players and. the machine ïras via a screen and. keyboard-

termlnal. Figure J belour illustrates the main operating

d.etaiLs of this prograrr.

FIGURE 3

ÏNTERACTIVE POKER SIMULATOR

All players stll-I 1n the game areall-owed to lmprove their hands. Thehuman players must type 1n which cardsthey wish to discard, and their re-placement cards are shown on thescreen.

44

The machine calculates the wlnner, andcomputes amounts won and lost by eachplayer. Then rne\^I game is commenced.

The, final round of bettlng takesplace, wlth aIl relevant bets dlsplay-ed on the screen. The slmulated play-erfs bets are computed by the program,wh1le the human players must key 1ntheir bets on cue from the sereen.

Players, in turn, nominate whetherthey wlsh to play, or not. Themachine 1s lnformed, vla the key-board, whleh players declde to ante,wh1le the actlons of the slmulatedplayers are descrlbed vla the screen.Doubling, if lt arlses, 1s handled inthe same vray.

Machine shuffl-es and deals the cardsto all players, where human PIaYershave their hands disPlaYed on thescreen. Speclal arrangements aremade so that each playerrs hand 1snot revealed to the other PlaYers.

START

tt5.

Because of the following practlcal difficulties

this work on interactive poker could not be caruieil to a

satisfactory conclusion (a realistic poker game involving

human players).

(t) Poker neecls to be playeiL with real money for the

results to have any practical signlflcaltce. This

wouLd. not only have been d.ifficult to arrarger but

it was felt that thls d.irection of research was

outsid.e the intend.ed- scope of the work, which

d.eals prlrnarily with method.s of find.ing optlmal

strategies for poker-like gameso

(Z) Poker ls always pJ.ayed. in sesslons ranging from

several hor:rs up to 12 hor¡rs or more. Ind.eed-,

good. players often vary thelr styLe of play perlod--

ically in ord-er to create uncertainty in the other

playersr and. this tactic is of great lmportanee 1n

the game of poker.

However, even if it ïrere posslble to overcome ob jectj-on (t),

46.

it was not possible to obtain the computer for the many

1ong, uninterrupted. sessions which would. have beên required.

in ord-er to oond.uct such an etçer1ment, as no interactive

tlme sharing system was avaj-1abLe.

Nevertheless, the results obtalned, wittr the inter-active poker simulator suggest that it wouLd. be possible

to continue work in thls area, using the interactlve poker

progran d.eveloped. here as a starting point.

2.1?_-=!¡0vçstigatlon of poker strategles þv simulatlon

It is theoretically possible to lnvestigate poker

strategies by slnulation in the following ïvay.

Flrst note that the strategy of each player in the

simulation mod.el presenteit here d-epend.s on the functions

r(n) and. g(t) (see eqs.1 and. 2), and. the arbitrarllychosen parameter e (see eq.3). Consid.er the parameter 8.

By keeplng the parameters (r(tt) , e(¿) ar,.d e) of

all players consta¡t, one player could be left free to vary

the parameter e as he chose. If a suffloiently large

number of games vuere simulated. for each d.ifferent choice of

e, he could. ileterrnine that e which gives him the þest

overall results under the given cond.itiorls.

However, thls approach has several d-isailvantages"

First, when a sinulation of this type was tnied. on a

simplifieil version of ttrls game (see chapter 4) it was

found. that a large number of games (several thousand)

\t.need.ed. to be elmulated. with any particular set of parameters

to d.etermlne the expeetation to any reasonable d-egree of

accUracy (".y to within 1% of the exact expectation which,

in thls case, could. ä1so be calculated. analyticdlly).

But since the game consid.ered. here is more complic-

ated than this, ard- since each hand. takes approxlmatelyI10

sêc. of c.B. tirne to slmulate, the tot al amount of c.po

time requlred. to cornpute each of these expectations would.

be high (several hund.red. second.s). Now, consj-der the

following ad.d.ltional factors.(i) This whol-e process would. need. to be repeated.,

in turn, fon each player.

(li) The strategi.es used. by each player are inter-

related. and. so (i) above would. need. to be

repeated until stable strategies ïr¡ere obtained.

for all players concurrently.

From the above analysls it becornes clear that this

method. wouLd. not be practicable (too much computer time

would. be required.), even if , as has þeen done here, the

poker progran was kept as slmple as possibl-e (i.u. þLuff-

ing was not introduced. and- only one variable, e, was

consid.ered. in the hypothetical experiment).

Thls failune d.oes not imply that a more sophigtic-

ated. approach to this problem might not d.rastically reduee

the computatlon time. But such work would. be eufficiently

48.

compLicated. to form a large, complete area of research,

and. 1t was d.ecid.ed. that thls Iay outsid.e the scope of thispresent vyork"

2¡ 1,Ã Conclud.inE_æearks

It was shoïrn that lnvestlgation of polter by simu-

lation is theoretically feasible, but, without furtherLreai,\¿\ .tl* *.il.*liç rLgu;UrÀ lr*,t,

d.eveloprnent, far too time consuming(u The program d.evelop-

eil hene red.uced- the c.po rurvring time by working out prob-

abillties and. hancL lmprovement in the fastest possible wB$.

Evid.ently, further progress 1n simulation must follow these

and. other lines if these method.s are to be at al-l practic-

able g Lsoo É¿^ÅL. [n,i'ù).

It became clear that the amount of work requlred. to

achieve effective progress in simulation would. be like1y to

be extensive. There was a choice of, continuing with the

research on simulation or of attacking the problem using

game theoretic methoils. After prelirninary lnvestlgation

it was reaLized. that lt wouId. be imposslble to do both,

and. thus the game theoretic approach vras chosen as itgeemed. likely to yield. useful results more quickly'

Findler,s pot<ir grolrp ls "r.""*ittl¡r vvssLing ined' out on simulationthis area, (37 ,38), and are obtaining so;ne

i¡teresting lesul-t.and. could. form a basis for further research.

Fina11y, the possibillty of using the poker eimu-

lator to play interactive poker was d.emonstrated.' A1-

though this id-ea has no d-oubt been consid.ered. before

49.

d.etalLs of e:çerimental work 1n this area have not pnevious-

ly been publlshed.. Interactlve poker ls an lmportant part

of poker sinulation slnce lt may be usecl to va]ld.ate ex-

perimentally the effectlvehèss of strategles calculated.

elther þy sinulation or by other method.e, by testlng them

und.er conclltions of actuaL play.

50.

CHAPTER 3

LOOKAHEAD (tAH) ALGORITHM

3.1 An Introducülon

It was shown ln chapter 2 th'at slmulatlon could not

be convenlently used to study the game of poker. An alten-natlve method of approaching thls problen ls to construct

mathematlcal poker models then solve these by game theoret-lc method.s. tühen thls approach was trled lt was found

that only the 2-person verslon of the partleular game form-

ulated could be solved by uslng standard methods (as found

ln the llteraturê). It becane evldent that more neal-

lsttc verslons of the game, because of thelr gneater com-

plexlty, could not be solved algebraloally, by avallablemethods, and thus numerlcal lteratlve methods were consld-

ered.

The only numerlcal method descrlbed ln the llterat-ure whlch ls relevant to thls study, [Rosen, (31)], 1s lter-atlve and requlres derlvatlves whlch cannot be obtalned forsome of the games treated here. lhus a modlfled"verslon of

Rosenrs method was suggested which does not requlne d.erlva-

tlves, but 1t was found to be slow for the Burposes of thlsresearch.

As a l"esu1t a new lteratlve method, called the

Lookahead (lnH) a}gorlthm, was formulated. Its speed of

executlon was shown to be at least 3 tlmes as fast ag

5r'

Rosents mettiûd r*hefl äþpl1ed to problems of the type that

are treated ln this etudyi

Thls chapten wlll be organlsed ln the followlng way.

A general statement of the problem w11I be glven. Next,

two numerLcal methods of solvlng games wtII be deserlbed.

The flnst of these ls Rosents method mentloned above, and

the second, the Lookahead (tAH) algorlthm, whlch ls the

maln theme of thls chapter.

3.2 Mathematlcal Statement of Problem

The games that arlse ln thls study faIl lnto the

categony of n-person non-cooperatlve games, and may be de-

flned ln the followlng vray. If, ln a game f , wlth n

players, each player acts purely ln hls own lndlvldual selflnterest, and coal-lt1ons between players are not allowed,

then I ls deflned to be a n-persorl nolt-cooperatlve game.

Conslder such a game I when the players are denoted by

the lntegens I = LrZr...n. The following deflnltlons

relatlng to thls game f w111 be used.

A strategy 1s the speclflcatlon of the courses of

actlon that a player w111 adopt 1n all glven sltuatlons

wlth whlch he can be presented. Let Sl, the strategy

space fon player' 1, be the set of aII posslble strategles

avallable to playen 1.

52'

Let crl e gt be some partlcular strategy being

used. by player i. When the n players are using

stnategies d! ,.. .cf,D Let d. = [qt ,...on ]T ba th" current

strategy vector. If S = S16l...@S¡ 1s the cartesian

¡lroiltrct of 31¡ o..9¡ then it follorvs that d. e S¡ Each

player i has a real valued. payoff functj.on fr(ø) which

assigns a unique payoff for every d€ S. Hence, ft(a),the payoff to player 1, is not only a functlon of hls own

strategy ct , but also d.epend.s on the strategies d,1 ,. . .crn

followed. by the other players.

The variable q. may be constrained. by p constraint

relatlonships of the form

e¡ (o) >

where the functions g1r...gp are real valued. functions

defined. over the set S. Fr¡rthenmore, each strateryctt €, Sr is ilefineiL to be a vector with fi1 real eompon-

ents cxt = lol ...oå,1T.

Eouilibrium points and. onti-nal strater¡ies

Nash¡ (25), introduced. the concept of the equilib-riun polnt (u.p.), whlch is commonly used. to characterlze

optimality in the general n-person non-cooperative ga¡ne.

Nashts d.efinition of optÍmal strategy ls applicable to the

games consid.ered. in thls study, and will therefore be used.,

anil is as follotva3-

53.

. Let dr t (i=1 ,.. orr) be an optlmal strategy forplayer i 1n the gane I, where cr = [otr...cn]T i" the

optimal joint strategy. Then Nash states that player imay not change his strategy crl wlthout suffering a de-

erease in his payoff functi on f r (ø) and. d.escribes thissituation as occurring at an equilibrium point (",p.).For example, if player i changes his strategy from crl

to o,r{ , the netu joint strategy 1s

q1, = loL ,az ¡. . .dto ,...on ]T,and. Nashf s d.efinltion of the ê.pr requires thatf r (a) >

It, at the end. of the game I, total gains equal

total lossee, then I is said. to be a zero sum game.

1.e. - 0 for all ct€ S.

rf = O for i=1 ¡...n and. i=1 ,...IIt1 thend

n> ft(q)

l=1

òfrã'T

Nash, (24), states that ttre joint strategy q. is an €.prS olving n-Er so n--ggg.es

For simple ganes, exact analytic solutions are

obtainable (see chapter l1). Hou¡ever, for more conplicateoga¡nes, numerlcal f terative method.s are requlred., as d.is-

cussed in the next section.

54.

L3 Rosenr s_ rLe.thod. of numerica] sol-gtion

Rosenr s method. approaches the ê.po by stepping inthe d.irection of increasing grad.ient (computed. fron the

d.erlvatives of each playerf s payoff function). In thisstud.y 1t ls not posslble to calcrrlate d.erivativeÉ and.

consequently the approximation

r, (x) ^ f (x+ax)_;--f (x-¡x), ax small2Lx

is uged.. Sinee this is the only significant change to thisalgorithm f\¡rther d.etails w111 not be given, but nay be

found. in (ry) ,

This algorlthm was able to solve the 2-person poker-

like game fornulateil. 1n chapter 4¡ þut it ïras slou¡ (see

3.9r. Since, later, [-person versions of the 2-person game

were to be treated-, speed. of execùtion v¡as eruclal-.a.lr The Lookahead. aleorlthm

The lookahead. (f,eU) algorithm was formulated. as an

alternative to Rosent s method. It is an iterative method.

that solves a game by approachlng the êopr in a series of

steps. When implernented. on the computen it was found. to be

3 tlmes faster than the nrocLif ied- version of Rosenrs method. 1

as used. here. The algonithun is based. on looking ahead. inthe sarne u¡ay that a chess player considers the replies open

to his opponent, before making his oÌÍD rnov€¡

The algorithm is first applieiL to a 2-person gane

and. shown to give âfr €.p. consistent tv:ith Nashts d.eftnltion.

55.This a-1gor1thm 1s then generalizeð. Lo the n-person case.

Finafly, two problems with lorovrrn solutions are solveil

correc tly.n.5 The LAH alsonithm i n a 2-DêFSorr oâme

Consid.er a 2-person non-cooperative game where

players 1 and. 2 control respectively the varj.abl-es qL and.

o.2, The joint strategy 1s d = føeraz]T and. the payoffs

f, (cr) and. f, (o) d.epend. f or each player both on hisstrategy and on that of his opponent. Suppose thatplayer 2 has fixed. qz and. player 1 is consid.ering a

change. in -d,1. Player t has no control over player 2.

Accord.ingly he consid.ers the effect of his opponentts

response first to a small- lncrement in dL, arrd then a

small decfement. Vfith thls knovyfed.ge he chooses the course

most likely to maximize his r:eturn. Player 2 respond.s tothe choice mad.e by player 1, in a similar way. The players

contlnue to al-ter their variables in turn until a stable

situation is reached. from whj.ch neither player is prepared

to d.eviate.

A simple numerical example will now be given toshow how this algorlthm works in practiceo Suppose in the

game d.escribed above the rules are such tllat 0 < qL <

O<o.z<

ft (c) = 5qL + 7qz 12q.Lq3

fr(a) = -ft(o).

56'

Assume that each player can only lncrement or

decrement hls varlable by t^ where A = 0.1, and hence

dl ro2 can only assume the discrete values 0i1r...

0.9, 1.0. Table 7 below glves all posslble payoffs fr(cr)

ln these clrcumstances.

TABLE 7

AN EXAMPLE OF THE LAH ALGORITHM

0.1

PLATER 2

vaLue of c2

o.2 0.3 o.¡+ 0.5 0.6 0.7 0.8 0.9 1.0

PIq¡er

1

value

of cr

0.1

0.2

0.3

0.h

0.5

o.6

0.7

0.8

0.9

1.0

1.08

1. h6

1.81+

2.22

2.60

2.98

336

3. Tl+

l+.rz

l+.50

L.66

L.g2

2. i-8

2.I+h

2.70

2.96

3.22

3.1+8

3.71+

l+.00

2.21+

2.38

2-52

2.66

2.80

2.91+

3.08

3.22

3.%

3. 50

2.82

2.8h

2.86

2. 88

2.90

2.92

2.911

2.96

2.98

3.00

3.1+0

3.30

3.20

3.10

3.00

2.90

2. B0

2.70

2.60

2.50

3.98

3.76

3.54

3.32

3.10

2.88

2.66

2.1+l+

2.22

2.00

l+.16

l+.22

3. 88

3. 5l+

3.20

2.86

2.52

2.18

1. Bl+

r-. 50

5. rh

l+.68

l+.Zz

3.76

3.30

2.81+

2.38

L.92

r. h6

1.00

5.72

5.11+

t+.56

3.98

3. ho

2.82

2.21+

r,66

1.08

'50

6.so

5,60

l+.go

\. eo

3.50

2.80

2.10

1. l+0

.70

-.00

Suppose that ln1t1ally cr=0.4 and q,2=0.3 , and

that each player ls aware of what the other ls dolng.

57.

Player 1 eonsiclers the folJ-owing J possibilities.

(r) Plaver 1 chooses a1=0.4

In this case player 2, knowing that ø1=O.4 will

attempt to maximlze his payoff fr(a) by changing dz to

O.2,0.3 or O,-4. Since player 2rs payoffs for these 3

choices are -2,llt, -2.66, and. -2.88 respectively, I u"

tr(q) - -f1 ( o) ] , he wi1] natur ally choo se d.2 =o .2 , as

this maximizes his payoff.

Hence lf pJayer 1 lets cr1=0.4, he nay ex¡reet

player 2 to chooge o.2=O.2. Thus the erc¡rected. payoff of

player 1 is 2.11)+. The algorithm ernploys only 1 d.egree of

look ahead- þecause it was e:r¡rerlmentally shorun that thls 1s

suff lcient for the algorithm to converge when d-eal1ng wlth

the class of ganes encountered. in this study.

(z) Player I choos es cl¿1=0,3

Using similar reasoning to that above player 1

calculates that if d1=0.3 r he may expect player 2 to

choose d2=O.2. Thus his elæected. payoff in this sltua-

tion 1s 2.18.

3) Player 1 choos es a1=o. !Player 1 computes h1s expected. payoff to be 2.7 in

this Gâsoe

Hence player 1 realizes that his þest strategtrr is to

choose s¿l-=O.5t as then his expected. payoff is maximized'

at 2,J.

58'

Now if is player 2r s turn to repeat this same

process. the starting point thls t ime is cr1=O¿ 5 and.

cx2=0.3. Ptayer 2 consid.ers the followlng J poesibil-j.tles.( t ) Playen 2 choos es cl2 =o. 3

In this case player 2 calculates that he may e:çect

player 1 to choose ø"L-Or6 (u" this maxlmizes player 1rs

payoff). Thus the expectect payoff for player 2 willbecome -2.94.(z\ Player 2 chooseg o.2=o.2

Now player 1 may be e:çected to choose ç¿1=O.6,

making player 2t s expected payoff -2.96.(Z) Plaver 2 chooses az=o.4

Again player 1 may be expected. to choose cN,l=Q.6,

making player 2t s expected payoff -2.92.Hence player 2r s best move 1s to choose ø2=Q.4,

as -2.92 > max[-2"94, -2.96, -2.9211. the so]-ution point

now becomes d1=0.5 and. d2=0.4.

In this way the solution point will keep moving

through the g'ame matrix until it reaches ø1=O.6 and.

cx2=0o4, from whlch point neither player is vuilling to

d.evlate, Now the grid. size A 1s d.lminiehed. by some

fraetion ø(O <

another stable polnt is reached-, The gr1d. slze is

again cLiminlshed., and. this process continues until A

becomes smaller than some pred.etermined value ê. ft

59.

will be shown in 3.6 that in certaln cases when the payoff

functions are sufficiently smooth, the flnal- point reacheil

will be conslstent with tl.e equilibrlum point of Nash.

Often problems in gane theory contain constraint

rel"atlonships, and. these are hand.led. as foLlows.

If at any tlme a varlable is altered. in such a way tTøt itno longæ satisfies the constraint relationships then itis moved. back toward.s its original value until the con-

straint relationship is again satisfied..

7.6 Eouivalence between T,ATÍ solution and. êor)r

In this section it w111 be shoïrn that for a 2

person game, where each player controls 1 varlable, and.

where the payoff functions are sufficiently snooth, there

is an equivalence between the êrp. of Nash, and- the¡solutionfound. using the I,AII algorlthm.

Consld.er the 2 person game d-efined in 3.5. Let the grid.

size A, used. 1n the IrAII algorlthm be very slnall, and.

assume that the payoff f\¡nctlons, irr the vicinity of some

point d = [øtrø2]T can be approxlrnated. by the planes

f, (ø) = ãLtrl.L + ã'zola + ârs (1)

fr(q) ! ãzrd1 + ãzzdz + ãzs Q)

which pass through the four grid- intersection points ad.jac-

ent to ø. then if f.(ø) and. fr(a) are sufficiently smooth

these approximations and their iLerivatives will converge

unffofftly to the f,r.i4ct1one f"(o) ahd

60.

f, (o) and their(see D¿C¡ Hand-scomb,correspond-lng d.erivât1veg, as

( 15)'l '

Hence ther d-erivatlves of the funotions fl(o) and.

fr(c) may be ctriosely appnoxirnated" Þy ttre d.erivatives ofthese planeer

Dif,ferentj.atlng eq.1 and. 2, and taklrìg the llnltas A-rO

òf,ãñ-ã'

A-rO

= 4ZZand.

A

ß)

In t,h1s sectlon some new notati on will be ueed..

Let

1 ,òfrãär'

S¡ (a1rø2) =òf'o;!frt=oòf';ããi.o

Thenefore from ee.J

âr r

I#Ti arr loS1 (ør ras) = (4)

O '¡ att = 0

Henee s ¡ (ar raz) 1s a constant, lnd.eperrd.ent of qL and. dB .

6t.

Suppose that the LAH a.l.gorl-thm Ís belnp¡ used on thls prob-

Iem, and 1t ls the turn of player I to move¡ The current

Jolnt strategy polnt 1s at E ln the dlagrán glven below.

Playen I must declde whether to remain where he ls at E,

or move r'lght to F, or move left to D.

(a -^, o2+A)

FC[0

I0 C t d , d

Thus he wtlI conslder the 3 sltuatlons (a), (b) and. (c)

glven be1ow.

(a) Player I remalns at E

Player I assumes that player 2 has 3 cholces, each

havlng the payoff as glven be1ow.

(1) Player 2 remains at E then hls payoff 1s

fz(ot rot) = àztdr+azzo2+azs(11) Player 2 moves to B then h1s payoff ls

f z ( clr ,c2+A) = âz ror+azz( a2+A)*azs

,

C[t

az)

G

,(t( (o,rfA,az)

H

az+a) (ot ,

D

o2+a) (o¡+4,

E

6z

(111) ÞIayen z meves to H thên hls payoff ls

fz(ol ro2-A¡ = âz rct'+^r"(o2-A)ta¿s

From thls lt can be seen that lf à22 > 0 then player 2

moves to B, 1f azz <

ând 1f ê-zz = 0 then player 2 remalns at F.

But as from eq.4

ãzz = 0 lmPlles that Sz = 0

and ãzz > 0 1mplles that Sz = Iand ãzz <

the above sltuatlon can be expressed thus.

Plaver I w111 exPect PlaYer 2 to

(1) remaln at E 1f Sz = 0

(11) move to B 1f S¿ = 1

(111) move to H 1f Sz = -I[hat 1s player 2 w111 move to s,2 + A.Sz

Player 1 moves to F

Uslng the same argument as ln (a) lt can be shown

thatplaver 2 w111

(b)

(c)

(1) remaln ab F 1f 52 = 0

(11) move to C lf 52 = I(fff) move to I 1f Sz = -]

That 1s player 2 moves to o'2 * A.S2

Player I moves to D

player 2 w1l-I

(1) remaln at D lf Sz = 0

(11) move to A 1f Sz = I(111) move to G 1f 52 = -]

61.That 1s player 2 moves to n2 + A.Sz

Hence, regardless of whether player 1 chooses ErF or

D 1t rnay be seen that player 2 w111 move from d2 toa2 + 4.S2.

Thrls, player 1 may now calculate h1s expected payoffs

1n the 3 situatlons

a) player 1 remalns aþ E(

expected payoff fr(cl ro2+4.S2) = ârrcrl+arzIe2+4.S2]*a¡3(b) player I moves to F

expected payoff fr (ar+4,o2+AS2) = arr(ol+A)+arzIa2+AS2]+ars

(c) player 1 moves to D

expected payoff fr(at-Aro'+AS2) = ar ¡(at-A)*arzIa2+^Sz]*ars.It can be seen from the above equatlons that player I

w111 maxlmise hls payoff fr (o) 1f he

remains at E lf âr r = 0

moves to F 1f ârr > 0

moves to D 1f â¡r < 0

It follows from eq. 4 tnat as 51 =

I i ârr > 0

0 i ârr = 0 then, to-1 i âtr <

maxlmlze hls return player 1 moves from ol to ol+A.S¡.

Hence, 1t has been shown that when A ls sufflclentlysmall player 1 w111 move 1n the dlrectlon of lncreaslnggradlent Sr, and w1Il only cease to move when Sr=0, that1s when the derlvatlve å# = 0.

64.

By a simlJ.ar means it can þe shol¡¡n that wtren A is

sufficiently small, player 2 will move in the d.irection of

his lncreasing grad-ient, S, and. will" only cease to noveòf.when Sz = O, that is ffi = O. thus, in t,lre case of a

2 person game, tuith each player controlling 1 varj-able,

and. the payoff fi:nctj-ons satisfying certain cond.ltions,

the solution found. using the LÆI algorithm will þe such

that the derivatives # and # equal zero at that

point. Hence this polnt will also be ârr ê.pr in the sense

of Nash.

By similar means it may be proved. that there is a

similar equivalence between the 2 solutions for the case

of an n-person game, with each player controlling arqr

finlte number of variables, provid.ed. that the payoff

functions fr (a) are sufficiently smooth. this will not

be proved. here as the proof used. is similar to the one

given above.

3 "7 Gene.rallsati-q1_-g|ls_elEorlthm þ the n-pergon case

lhe LAII algorithrn can be generalised- to the

situation d-escribed. ln 3.2, involving n players, where

player i controls rn1 varj-ables.

Suppose that the solution point is currently at

some point c[ = [c,ú1 ,...dn]T and. player i, who controls

the variabl-es ql ,...o;, must d.etermine whether or not toal-ter any of h1e variables. Player i will treat each

65.

varlable ql, . . .o,å , in the followlng vyay. First he will

consid er a! and- he wifl compute h1s expecteit payoff for

the J possibllities,

(i) oQ altered to aj+A

( ii) al alterecl to ql,-A

(i-i:.) "l remains unchanged..

He then chooses that value of al which promises him the

hlghest expected. payoff, which is calculated. in the manner

d-escri-bed. beIow.

The e:qpe cted. payoff to player i for each of the

above J possible choices of al is calculatect as follows.

l,et each o f t he other n-1 playens be consid.erecL in turn,

and. al1ow each one the option of temporarlly changing arìy

of the vaniables und.er his control by an amount lA, in

such. a ïvay that his own lndividual payoff is maximized,.

When this process has been completed. fon al-l players the

joint strategy d is changeil to some d{, ard. the

expected payoff of player i is given, for that particular

initial cholce of oL, by fr (aü).

Player i repeats this process for a1l- his other

variables qL r.. octj ,.fn thls way, starting with player 1, and. finlshing

with player n, each player is given the opportunity of

changing his variables, and thls proced.ure constitutes one

cycle of the algorlthm.

66.

These cyeles are eoi:ti-nued. untj.l- -ihe ioint strategy

d. remains unchanged. for 1 cycle. At this point the grid.

size A is made smaller by some factor r, and- the

complete process is repeated. with the neïu A. The

algorithrr stops when A becomes snaller than some pre-

d.etermined. minimum grid size e.

A flow chart of thie algorithrn is given 1n figure l¿.

7"8 Aoolications of the I.,Æi alEorithm

Three applications of the LAII algorithrn to problems

with known solutions were consld.ered. In eadr. case the

correct resul-t was obtained.. Two of these problems were

Linear programs reformulated. as games, (n) r arxl wilL not

be d.iscussed. here.

von Neumannls po_ker- Like -garne

The rules of this game have alread.y been glven in

chapter 1.

The game is solved. in the following nrâ$. this

solution w111 be given in some deta1l as the method.s used.

play an lmportant role in the ganes solved. in clr,apters l+

and 5,

It is assumed. that player '1 folLows a strategy

governed. by the function Øt(*), where Ør(*) is the

probabillty of making the largen bet, that is A units,

for a given hand. xr

von Neumann shorus that ttre n.lIes for this game lead.

to the functlon Ør(*) having the general form given below,

6'lFIOURE ¡I

FLOH DIAORIT¡I OF LAHALOORIIHÈ|

haé beon ob

Êtop: a Eolutlon

has minlmm Êtep s1z6

e been reachô¿

ô+nÁstep elze À

d@ê tho nof, €olutlon IequÀ1 the old eotutlon é

thus, each of th€ , v31ues of tT slve ¡ value of fD(g)loplaco rl by that choico f,h1ch glvos tlte higheEt fp(g)

not

¡or conalaler, ln tum, each component c|(klI))

alrs-Àaleøclee+Á

anil flB]ly replaoE dT

whlch gives the Mxlmhby thåt cholcê

for the th¡eê cholcegÊ ê d, md calculatc

Ln fùo folloYlng way.lat¡¡ (s)

Ilrg

each of the three cl¡olce. of

"?-t-atÏ -t

"l*t+¡1n the followlng way

rd conslderlet t+

of1-th varlabLe

1 - l,,..n!

conEialer the p-th plEye"D - 1,...n

C*¡Ebè latest solutlon tna¡ray Ê

lnltlallze vå¡labIesn + nunbêr of player!ü + orlglnal Jolnt Etratêt¡r vscto¡A * lnlttal 6tep slzee + nLnlnum step slzed + ratlo between EucqêEB1ve ltap slzasnpê nunb.r of variabled controll.d by plsycr p

START

rolreat alSorlt¡Drl.th tàc a[auo!

st€Ir BLzg

ye6

68.

and. this form of ttre fcnct,i-c'-l I s 'rset her-e"

^

I

0r(x)or x>b

0 a b I

The above functÍon, 0r(x), descrÍbes a strätegy

ln whlch player I bets A unlts 1f hls hand 1s elther

very good, (x>b) or very bad (x.a). The latten bet

on a low hand corresponds to the bluff. Ïf the hand x

1s such that a<x<b then player I bets B unlts.

The strategy for player 2 1s Blven by the functlon

ûr(y), where Vr(y) 1s the probablllty that player 2 looks

at a bet of A unlts by player 1, lf he (player 2) holds

hand y. Fon slmllar reasons to that glven above 1t 1s

assumed that Vr(y) ls of the form

^ür(Y)

¿

x

c0 1y

t

69.This means that p1âyer 2 on.-tJ'looks at a bet of A unitsif his hand. is reâsonably good.. Note that there is no

opportunity for player Z to bluff in this situation.ïf both players follow the above stnategies it is

possibJ-e to calcul-ate the expected payoffs to players1 and. 2 as f,unctions f"("rbrc) and. fr(arbrc) nespective-lyr and- as this i-s a zero sum game, (1.e. the losses ofone player constitute the gains of the other)

f2 ( arb rc) - -f1 ( arb, c) .

Before calculating the payoff fì¡nctions a new term

must be def1ned..

w; if rt>y

xU'¿ (*ry)¿ if x<yt

Now it is possibLe to list arr the Bossible playsr.cornputetheir probability of occurring, and. calculate the corres-pond.ing payoff to player 1 in the following ïrâJf. Consid.er

the play BB (i.u. player l bets B, and player Z bets B).The probability of player 1 beËting B is

f1'Øt(*)]' and. the conditlonar probability of player z

bettlng B 1s 1 (as player 2 is forced to bet B when

player 1 bets B).

Therefore the probability of this play is11-Ø r(*) ]vt , and the payoff to player 1 is BXt,- r (*,y)because if x)¡rr then player 1 wins B units

70'Q|''t(xry) = *1 for *ty) otherwise if 2?s hand. wins,

1. e. {)xt then player 1 1os es B units (Xa r- t (xry) - -1

for x<y). The event x=l of a d.raw has probability zero

ar:d. so d-oes not affect these calculations.In this manner the 3 different possible plays with

their pnobabillties and. thelr payoffs may be calculated.

anit are sunmarised. in table 8.

T¿BLE B ¡ SUMMâRY OF PLAYS T'CR VON 1 S GAME

B

B

A

Xr,-t(xry)2

xL '- t(xry)2

1-Ø r(x)Ør(*)11-Qr(v) lØ

'(*)úrfu)

B

A

A

B

B

A

Pl-av

rIProbabill tv Payoff to I

I

Hence fon æry given hands x and. yt dE, thee:r¡rected. payoff to player 1 is given by multiplying thepayoffs for each different pray by the probability of thatplay odc\rnring and summlng over all possible playse i.e,

dE=811'*'(x,y) .11-ø, (*) l+W, (*) L1-út(v) l*&å ,-L(x,v)ØtG)/ r(v)

The total erqpected payoff , f r(arb ,c), 1s obtained. byintegrating d.E over the range O<x<1 and. 0<y<1 , (fora slmilar calculation see pnritt, (Zg).)Thus

TT.

f1(arbrc) = lv(t-ør(")xl.- 1(x,y)

x=O y=O

+BØ¡(*) ( t-Vt(y))+¿dr.(*) úr(v)x'";-'(xry) i¿* ¿v

I I1

1

II

x=O y=O

1

+

I

I

I

ø(l -ø r(*) )tt, - t (x,y)clx dy2

+ B'Ø *(x) ( 1 -þ "$) ) ax ov

X=0 Jr=O

+ | | Mr(x)tt(v)xl,-1(x,y)dxdy.I I

'*r "4x=0 y=0

Nor by using the definitions of Ør(x) and. úrß)given earlier, 1t is posslble to expand. tTre above integrals

as follows..b

^1fr(a,b,c) = I I B^¡t,-t(xry)dx dy

X=å' l/=Q

lB l'nd.xcty + I

1

:r=b ¡r=O

dx dV

x=O ¡r=O

.à .1 .1 ^1.J l¡xLz-'(x,y)dx'd.y+

I lo*rr-'(x,'v)dxdyx=O JI=G x=b Jr=G

f.("rbrc) = BIr+Bac+B(1-b)c+ArsrArs whereThus

72.

TL.L rlX=â ü=O

rt1

xt,- " (*ry) dx dy

TL2 Xt,' " (*ry)dx dy

X=0 ¡t=C

I 1

Is= (*,y)dx dy

X=b y=g

This integral presents some dlfficulty in evalua-

tion, and. occurs again in chapters 5 and 6 1n a more com-

plicated. form. Because of this a general ïyay ïuas found.

of evaluating this type of integral, and. 1s given 1n

Append.ix A, The nesults obtalned. in append.ix A permit

f lrIz and. Is to be evaluated., and. it is found that:11 = (¡-") (a+b-1 )

-a(t.'c) i a< c

.-a(j-c)+(a-c)'; a> c(t-"¡(t-"); b <

(t-¡)(¡-"); b > c

I I

r2

rs

=[

=[

îhe constrain'u relationslrips governing th-e ya"rlabLes

ârb and. c are:

a < b, and arbrce [Ort].ïyith the above information the problem was solved. by means

of the LAII algorithm, which gave

a = O.'l r b - O.7, c = O.l+.

These results were consistent with those glven by

73.

von Neumann irr (n). By evaluating certain d-erlvatives at

the €opn it may 'be shotlr¡: that this optirnal strategy aLso

satisfies Nashr e cond.ition for ârr €.pr

3.9 Comnanison between Rosent s mod.ified- method. and. the

I,ÆI alsonlth¡n

A 2-person poker-like game 1s formulated. inctrapter l+ whlch is typical of the claes of games to be

treated. in this stud.y. This 2-person game has a knovrn

exact solutlon, given in terms of variables arbrd. and. er

The meanings of the variables are unlmportant in the

present discusslon and. are therefore not defined.. Several,

comparisons of the mod.if ied- Rosent s methoiL and. the LÆI

algorithm, as applieÖ to thls game, ulere mad.e. lable 9

given below shows the example that was most favourabl,e to

the mod.lfiecL Rosenrs method..

TABLE 9: COMPARISON BETVTEEN ROSENIS MODIFIED MEIITOD Æ{D LAII ALEORITHI{

It can be clearly seen from the above table that the

LAH'a1gorlthm, even 1n thls lnstance ln whlch lts perform-

ance v¡as least good, achieves a more accurate solutlon

30

9l+

.81056

.81056

,BL9T3

.6tz6T

.6tz6g

.6rhl+B

.6t+r77

.6t+L79

,6577o

.02980

.02980

.02803

exact sol-n.

LAII alg.

Rosen I s motllfÍed. method.

time( c.D. second.s )tTcba

7tt 'than Rosenrs modifled method ln only ! of tne tlme.

J

For thls reason the LAH algorithm ls to be preferred

for solvlng the class of games dealt wlth ln thls study.

75.

CHAPTER 4

F'ORMULATION AND SOLUTION OF A 2-PERSON POKER-LIKE GAME

4.1 Introductlon

Chapter 4 ls concerned wlth solvlng a 2-person

poker-llke game (2-PG). The more dlfflcult task of solvlng

the 3 and 4 person verslons of thls game 1s left to chapter

5. Thls work has some slgnlflcant features whlch are

mentloned below.

(a) A search of the llterature suggests that thls ls the

flrst tlme that solutlons to a poker-Ilke game have

been sufflclently rea}lstlc to predlct strategles

commonly used by experlenced players.

(b) The solutlon found to thls poker-1l,ke game 1s app11c-

able not only to the analysls of poker, but also to

certaln types of buslness sltuatlons. Furthermore a

network problem |s solved by the methods developed ln

thls stud.y to obtaln a new and lnterestlng result.

These two toplcs wlIl be d,Lscussed ln chapter 6.

Before proceedlng to deflne and solve the 2-person

poker-llke game (2-PG), the orlglnal poker game on whlch tþe

z-PG ls based w111 be dlscussed. From the nules glven ln

chapter 3 1t may be seen that thls poker game nay be conven-

lently dlvlded lnto the 2 phases glven below (famlllarlty

wlth the poker game descrlbed ln chapter 2 w1L1 be assumed

1n the remalnder of thls chapten)

76.

Phase 1 of the poker game

Phase I of the poker game extends from the deal up to the

lmprovement of handso whlch ls achleved by dlscardlng un-

wanted eards and replaclng them wlth new ones from the deck,

Phase 2 of the poker game

Phase 2 of the poker game conslsts of a round of bettlng,

after whlch the'wlnner ls determlned.

A relatlonshlp between phase I of the game and the

entlre game w111 now be establlshed.

Conslder the above poker game under the slmp11fy1ng

assumptlon that no further bettlng ls to be alLowed ln phase

2, and call thls the phase I game. Experlenced players

assume that a sound optlmal strategy for the entlre game

must be based on a sound optlmal stnategy for the phase Igame (see (30)). Thls 1s Justlfled 1n the followl-ng way.

Observatlons of games played by experlenced players

have shown that nnosl {ãc: ker hands have no s1gn1f-

lcant bettlng 1n phase 2. Thus, th these cases, the game

reduces essentlally to the phase 1 game and hence the phase

I optlmal strategy ean he used. Now eonslder what w111

happen when a player uses the phase 1 strategy

i)r -' .,,r,, ,',' '' ': where there 1s slgnlflcant bettlng ln

phase 2.

Books on the subJect (8130), lndlcate that ln thls

case the strategy followed 1-n phase I ls overshadowed by

77.the strategy foltowed ln phase 2, That 1s the elements of

blufflng and poker psychology are paramount, and, to a large

extent, the exact strategy followed ln phase I loses 1ts

lmportance. The use of the phase 1 optilnal strategy there-

fore, canrles wlth 1t no slgnlflcant penalty when applled to

Iks cors¿ n -, ',, '! For thls

reason the phase 1 optlmal strategy ca-rrr, be used ln the

entlre game (whlch conslsts of phase t followed by phase 2),

otherwlse losses w1ll be sustalned 1n the

lnstances mentloned above, where there 1s no slgnlflcant

bettlng 1n phase 2.

The complexlty of the game 1s such that 1ts entlre

solutlon was consldered lnfeas1ble, partly because of the

dlfflculty of descrlblng the payoff functlons mathematlcally,

and partly because of the dlfflculty of flndlng an e.p.

Therefore, the phase I game only 1s solved 1n thls

study, and thls solutlon is then used. to formulate a strategy

for the entlre game ln aceord wlth the assumptlon made

earller,Presentatlon of materlal

In thls chapten a 2-person verslon of the phase Igame w111 be deflned and solved by means of the LAH algorlthm

(see chapter 3).

The 2-PG ls based on the game descrlbed 1n chapten

2 Sectlon 5 (of whlch the relevant parts, for convenlence,

78'

are repeated here 1n shortened form), but has slgnlflcantdetalled dlffer"ences âs follotus.

The game 1s played by two persons called player Iand player 2. Both players recelve a hand of cards, where

the hands are respectlvely represented by random numbers

x and V¡ that are unlformly dlstrlbuted over the closed

lnterval [0rI] (see 4.5). The play befone hand lmprove-

ment ls summarlzed 1n the flow dlagram glven 1n flgure 5,

The hand lmprovement 1n thls game ls based on certaln

slmpllfylng assumptlons whlch are made by expert players ln

order to analyze the hand lmpnovement process (B),

and are glven below.

a) The flrst assumptlon 1s that each player 1nlt1aIIy(

holds exaetly I palr

The Justlflcatlon for thls assmptlon ls that players

seldomr lf ever, play on less than I palrr(30). It may be

established from table 2 ln chapter 2, that lf hands weaker

than I palr are ignored, then 7 out of B hands are I palr.

Thus assumptlon (a) correctly represents real poker 87/'ofthe tlme. But slnce the probablllty of lmprovlng a palr

ls 0.287 (see table 2, chapter 2), 1f (a) above ls assumed,

1t follows that the probablllty of lmprovlng any hand 1s a

game constant equal to 0.287. AIso, slnce from (a) aþove

all hands are lnit1ally pAlrs, lt follows that lf one hand

lmproves whlle the. other does not, then the lmproved hand

must wln.

FIGURE 5

FLOII AGRAM OF 2-PG

79

player 2 bet"s 2

PlqYer Ia,nte 2 r:nits

cr¡nulative bets ofplayer 2=2

player I bete 2cr:mul-ative bets of pleyer_l

-¿

player 2 bets l+

pra.yer 2 antes a--- :ffr#l}"rlit"further 2 r¡nits2

retrieves Idrops out of the ga.me

pfs¡¡er 2pleye on

after simul-atecl hancllnprovenent ninneris cleternined

(:. . e. cloubles\l

plelfer I 1 bets 2plsys on a¡al player t\ cunulative bets

antes 2 units clrops out of PlaYer 1=l+

player 2

antes 2 r¡nlts

Bleyer Ihas 2 choíces

Pla,fer, ?

retrleveF hls ante &

gamç encls

player 2

has 3 choices

wins

Df8^yer Ihas 2 choÍces

v].na

80.

(b) The second assumptlon Ls that If 2 hands lmprove then

the hleher orlglnaL hand w1ll wln

Fnom (a) above the probablllty that 2 hands lmprove

ls (O.aB7)2 n: 0.08. Slnce, lt 1s tnue that 1n thls case

the hlgher orlglnal hand wltl wln at least half the tlme,

lt follows that the probablllty that assumptlon ( b) 1s

lncorrect (f.e. the lower orlginal hand wlns), ls less than

ä(0.287)2 = O.O4 (\%). As a consequence of thls 1t follows

that ( b) 1s coryect ln better than 96% of cases.

The probablllty of all the above assumptlons belng

correct concurrently (obtalned by nultlplylng condltlonalprobabllltles) ts 0.87 x 0.96 = 0.84 (8\/"). That 1s the

assumptlons break down ln ßf" of lnstances.

Hence, lt would be reasonable to follow a strategy

based on these assumptlons provlded that heavy losses are

not sustalned ln the L6% of lnstances where at least one ofthese assumptlons was wrong. But lt has al-ready been men-

tloned that ln at least 85% of all hands there 1s no s1gn1f:

lcant bettlng 1n phase 2, and ln thls case no more can be

lost on the 16% of occaslons when the assumptlons do not

hold, than ln the B4f' of lnstances when they do. Further-

more, the probablllty of assumptlons (a) and (¡) belng wrong,

and there belng slgniflcant bettlng ln phase 2 1s

less than .16 x 0.15 = 0.024, and as noted 1n 4.1, the aet-

ual strategy and assumptlons used ln phase I 1n thls lpstance

are unlmportant 1n any ease.

8r.As a consequence of th1s, the experlenced player

flnds lt reasonable to use the slmpl1fylng assumptlons (a)

and (b) (see (30)) whlch are cor.rect ln 84/" of cases, slnce

he knows that ln the L6% of lnstances where he 1s wrong, hê

w111 not sustaln any heavy losses as a result of belng

lncorrect.

Mathematl cal deflnltlon of aonroxlmate hand lmprovement

In accord,ance wlth assumptlons (a) and (b)

made earl1er, the mathematlcal deflnltlon of hand lmprove-

ment 1s made 1n the followlng way.

Deflne the game constant g¡ where q = 0'287'

Then Tq ls a random transformatlon defined as

Tn(z)z

2tz

: wlth probablllty l-q

: wlth probabllltY q(r)

1s replaced by

Player I wlns the game

At the end of the 2-PG, ,c

X=Tn(x) and. y by t=fn(V).lf X>Y andptayer2 if Y>X.U.2 StrateEles of players

The strategles of the 2 players are descrlbed by

the functlons 0r(x), 0z(x), Vr(y) an¿ Úz(y). The mean-

lngs asslgned. to these functlons are dlsplayed 1n graphlcal

form ln flgure 6 below.

{¡ops wlthprobab11ltyI - 0r(x)

game ends

plays wlthprobablllty

ûr(Y)

PLAYER 1

drops for halfwl.th probablllty

I - ,l,r (y) - tz (v)

FIGUFE 6

STRATEGIES FOR THE 2-PG

anbes wlthprobabll-1ty

0r(x)

PLAYER 2

82

doubles wlthprobablllty

úz(Y)

PLAYER 1

plays wlthprobablllty

0z(x)

dnops wlthprobablllty1 - 0¿(x)

hands arecomparedto determlnethe wlnner

endspla¡rer 2

wlns

83.

Flgure 7 glves the form of the strategy funetions

and 1s to be lnterpreted ln the fo]lowlng way. Consider'

for example, the functlon 0¡(x). Thls means that player I

only ever makes an ante lf hls hand x ls less than

(a bluff ), or greater than b. He wll'I never play on a

hand between a and b.

observatlon of experlenced players as recorded ln

(30), an¿ solutlons to other poker-llke games , (27), suggest

that (1n addltlon to the matters noted above) tne strategy

functlons have the followlng propertles:-

(1)Playerl].ooksatadoubleonlylfhlshandxlssuchthat x> c.

(11) P1ayer 2 always plays on 1f h1s hand 1s better than

d.Hedoublesonhandsbetterthane,orlessthanf (tfre latter bet belng a bluff). Otherwise he drops

out.

Theformofthefunetlonschosenlmpllesthatp}ay-ers never follow a varlable strategy, 1.ê. one ln whlch the

probabllitles of certaln actlons Ile between 0 and I.

Thls 1s substantlat,ed ln practlce, and ls explalned by

Bellman and Blackwell (2), ln the followlng way.

84.

1

0r(x)

0

0z(x)

0

I

ûr(v)

o

üz(v)

I

FIGURE 7

2-PG STRATEGY FTINCTÏONS

PLAYER I: has a hantl x

a

I

PTAYER II: has a har¡d y

cl e

probablJ.ity of pl4rer I naking an

ante for a girren ha.ncl x, where

L 0 < a s l, 0 3 b < I ancl a I b.

probabiliff of ptayer 1 looking

at a d.ouble for a given hantl xvhere 0lcl1.

probabitity of p1ryer 2 PlaYinga given ha¡rcl Ír where 0 < d < I'0<e51 entl cl 3e.

probabil-Íty of plaYer 2 doubl'fng

for a glven hantl y¡ where

0 < f < 1, 0 I e I I antÌ f r d.

c

I

II

I0f e

85.rrln a contlnuous game, one alIows the game, whlch furnlsheS

a randorn card, to do the blufflng. It turns out that 1f

the cards are dealt at random, any further randomizatlon

furnlshed by ml-xed strategies on the part of the players

ls superfluous.rl

Thus, thls approach uses a very broad general

observatlon, made by experlenced playersr based on sound

practlcal assumptlons, to flnd a solutlon ln much flner

detall. The results flnally obtalned show that thls

appnoach ylelds satisfactory results.

4.3 Evaluâtlon of oavoff functlons and solutlon

The payoff functlons for the 2-PG are evaluated ln

the same way as the payoff functlons for von Neumannrs game.

Flrst deflne the functlon

*ä;*(x,y) = the expected wlnnlngs of player I (2)

holdlng hand x, comPetlng 1n the

z-PG agalnst player 2 holdlng hand

y t where plaYer I elthen stands to

galn w units lf ln(x) > Tq(y) or

L unlts lf Tn(x¡ < ln(V),

ána, as 1n chapter 3, deflne

*:" ( *,v);x>y; x <yI

then from eqs. 1 and 2 tt follows that

(3)

x''l(*ry) = [q2 L

prob. that both PlaYens imProve

+ prob. that nelthen player improves

86.

l[ïï iN(4)

+

+

prob" that player 1 imProves whileplayer 2 does not

P1ob. that PlaYer 2 imProves whileplayer 1 d.oes not

o'W

.¿

Now, slnce from êQ. Iprob. that a player improves = q. (5-

ancl hence

prob. that a player fails to improve = 1-q ('6)

it follows that by using eqs. 3, ! and 6 , €8' 4 may be

written

xn,2 (*,v) : Iq.' + (r-q)'JxT'¿ (*,Y)q23

+ q.( 1-q.) vr

+ (t-q-)q.¿.

Hencer oñ simplifying

*i;t (*,v) = [q'+( 1-q)" l*i" (*,v)+(w+l ) q-(q-q) (7)

Now it is posslble to proceed. as in chaptet 3.

Table 1O befow gives the foLlowing information.

(t) All posslble outcomes which may occur in the gamei

(¡) The probability of any particular outcome occunring for

87

TABI,,E IO

SUMMARY OF PtAY FOR THE z-PG

Prob 11lt o aGame outcome rou c

(the superscrÍpts belor re-fer to the Prqrer naklng apartlcr:.lar nove)

play1.Dropz Øt(*) .lt-þt(y)-Ún(v) l

P1ay1.Playz Ør(*) þr&)Playl.DouþIea "Dropl Ør(*) Ú"fu) . Íj4r(*) l

P1ayl.Doulclez.Play1 Ø r,(*) Ùr y) Ø r(x)

+1

*3;-' (x,Y)

-2

xL '- t (*rY)q2

88.

a glven set of hand.s x and. fr (flrese probabillties

rnay be eas11y fou¡ril by referring to the mles of the

game anil the frrnction d.efinitions glven in flgure 7).

(") The payoff occurring to player 1 for a partlcular game

outcome (as d.efined. by the nrles).

Now, aS in chapter 3, Pt, the total expectecl payoff

to player I, is given þY

.1 .1P1 = t Ilø,(*)t1-t,k)-ú,(v)] (B)

JJx=O Y=Q

+ Ø t(x) þ t(v)x', - z (x,v)

-2Øt(x)ú, (y) l14r(*) l

+Ø "(x)

ú zG)ø, ( *)xl L-

r (x,r) l¿*¿v

As this is a zeto sum game , Pz, the expectecl payoff of

player 2 is -Pr.Hence Pz = -Pr. (9)

In orcl-er to evaluate eQ. B d.efine

Ø"(x) = Øtk)ør(*) (10)

aniL

Ør(*) =Øt(*)[r -øz(*)1. (tr¡

Consld.er the functi-on ø"(x) glven by eq. 10 aþove.

Figure 7 d.efines ttre functlons Ør(*) and. Ø"(*) in terms

of varlables ãrþ arrd. c, where a ( b.

Then 1t is simple, though ted.1ous, to shou' that the

function Ø"(x) has the form

89.

A

v

wfiere

and

g a

1

1

xh I

g = mln(arc)

h = max(brc)

(t2)( 13)

By s1mllar means lt can be shown that 0,'(x) fias

the form

^t

0r(x)

where g and h are as deftned by eqg. 12 and L3.

Thls nesult may be stated algebralcally as

oror

oror

gI

Ø"(x)=[î;3:i:å

ØoG) = [î ; å: i::

a(x(hh<x<1

h<b<

and.

90.

Now eq, B nay be wrltten

I1

ÎtL¿L

X=O y=O

^1

I Ø,G) 11-ü,k)-t"(v) J¿*¿v

Ø r(*)þt(v)xi ,'2 (x,v)dxdv

1

Ø, (x) ú"fu) ctxcty

Ø" (x) úrk)xf,'-a (x,v)dxdv

XÀ , -. (*ry) d-xcly + *t";'o (*,Y)d'xdY

1 1

+

x=O y=O

-, 1 Iic=o y=O

I 1

I I

+

x=O y=Q

and. by using the d-efinitlons of the ftrnctions

Ør(x), Øz(x), øt(x), Ø+(x), Ütk), Ù"fu), '

and e:cpanùing the 1ntegra1, arrct evaluatlng some of the

component suþ-integrals, it can be shown that

P1 =(r*1-b)(¿-r)¡ã ¡€ el r'e

+ f [-x',-'(xry)dxdY- I I t2t-'(xrY)ttxdYJ I

' \^t.v¡lruÃs')' ' I I ^o,x=O Í=d x=b y=d.

H](=$ }r=€ x=h y=ê

I I

+ rII

a

1

It *',-¿ (x,y)axay . I f .'"'-¿ (x,v)dxdv (ro )+

x=g $-O X=h Y= Q

9r.

where

and.

Now d.efine

g - mrn(arc)

h = râx(1, c) .

XQ,ä, '(;b342

" *i;t(*,Y)d'xclY

anct uslng ê8. 7

)(=at Y=àZ

J(=41 Y=àZ

xq''(^: å:) = f' f"ø*')q(r-q)d*dv

2I q.'+ ( t -q.)' l*\" (*,v) clxov+

)(=al Y=ãZ

Xä b2d2

X=â1 y=dz

where it has þeen shown in append.ix A that

and. as in chapter , d.efine

with

f' f" *:"(*,v)ctxdv

.. Þ r*üI + w(¡"-u) (-/ \2

,¿ (b,\*" )

( rz)

( ra)

. ,ba+vI - w(r,-v) (-/ \2

ã2

?2

ãp)

( rg)

and.

[ = flâx( a1 ,nin(b' a, ) )

d2

92.

v = tllâx( a, ,nln(b1 ,þP ) ) .

Hence by cornbinlng eqs. ]-7, L8 and L9

xQä ' t G: k) = q-( r -q-) (w+r ) (b,-', ) (t,-a, )+

lq'+(1-d'lxy'¿ lbt\.' &2

't

( zo)b2

Now eq..16 may be written

P1 = (a+1-b) (¿-r) 2(g+h-t) (t-e+r)

+ xQã ,-"(l

(2r)

* XQf'*z

ag

f\a)

+ (ve'|' 1

e) . **'-'(l 1)

+ xQå,-.(Ë å) . xQå,-.(l 3)

where g = mln(arc)' h - max(trc)

and. xQË,'Gl k) rnay be found fron eq. 20 .

Before tTre problen may be solveil by the LÆÏ

algonlth, the constralntg on the variables ãtþtcrd.re and.

f must þe speclfled.. Consld.er tfre d.eflnitions of Ør(*),

Ør(*), út$), úrG) as glven ln flgura 7. It rnay be

notecl from tþis that the following constralnts hold.t

a(bd<e

and. ârbrcrdrerf all lle in the closed. interval [0rt].Qz)

93.

uslng ttre aþove clata Ít 1s now poeslble to eolve

the z-Fle using the T.,AÌI algorithn, for differlng values of

Qr rn each case the startlnS point of the lteratlon is

zero, ffid the nlnlmum step slze ls 10-6. A liet of the

€,p,rs obtained- is glven in table 11 þe1cw.

x'ron the definitions of the strategy functlons

(see flgures 6 ard 7) it may be seen that¡

( t ) c is the mlnimum hand. requireil by player 1 to

play on (foo]r) after a d.ouþle by playen 2i

(Z) player 1 never plays on a hand þetween a anÖ b'

Thus, from (t) an¿ (Z) aÞove, lt follows that the

exact value of c 1s arbitrary if a < c < þ, even ttrough

in the solutions given above c=b¡

h.¿r tic s t ion of ttre qâme

L.¿r.1 Initial assumptlons

The method of solving the game analytically is a

2 stage process, fn the first stage, to be given in this

sectlon, certain aSzumptionS a?e mad.e, and. conseq.uentl'y a

solution is oþtained. The second. stage of thls proceetst

given in l1.l¡.2, valid.ates these lnitial assumptloIIS¡

sectlon 4.ft.J will then show that the z-Pe could. not be

solved. ana1yt1ca1ly if an approximate numerical solutlon

was not already known.

In the previous section 1t has been shown that the

solution to this game, when g = 0.287 le:

94.

TABIE 1I

SOLUTION THE 2-

0.17?8

0.181+2

0,189?

o.1gb0

0.2000

0.0000

0.0000

0.0000

0.0000

0.0000

o.æ6'l

0.81+38

0.8236

o.8to6

0.8000

o.6667

O.6l+2,

o.62ht

o.6l;2'l

0.6000

0.7333

0.?035

0.6?08

o.6ht8

0.610?

0.?333

o.?035

o.6T08

0.6|tr8

o.6to?

0.0889

0.0?r9

0.0502

0.0298

o.oo?6

0.0

0.1

o.2

0.28?

o.l+

pqroffto

plryert

îedlcbaq

95.

â.=.o2g8r 'þ=.61+18r c=.6418, ô= '6127r €='8'106'

f = o.oooo (z¡)

Suppoee that, rnotlvated by the above aBproxlnate numerlcal

eolutlon, tlre following lneguallty relatlonshlps are

assuned. to hoLd between the varlableS at the exact êrpo

â(dr a<e' d<b<e (24)

The correctness of thls assunptlon will be sholflr

ln 4.1+.2.

Now fron eq.24 it nay be establlshed that

a = max(ormin(ard))

b = &âx(t rrnfn( 1 ,d) )

¿ = nax(arrnln(are))

. ê = nax(arnin(1re))

e = nax(b rmln( 1 r e) )

a = max(armln(arO))

b = ßâx(trmin(1ro))

and by usLng êQsr lg r?O r?l and25, after some lengtl¡y

algebra lt nay be shown that Pt reduces to

Pr. = (¿-r) ( t+a-t)-za( t+r-e)

+zQ[ ( e-d) ( t-e-a)+(e-t) (b-ð+ze- 2) +2f (1 -b) J

where Q=q2+(t-q)'and thus fron eq.9

Pz = -(¿-r) ( 1+a-t)+za(1+f-e)

-?e[ (e-d) ( 1-e-a)+ (e-t) (b-d+2e- 2) +21 (1 -b) I

( z¡)

þø)

Qtl

(za )

96.

It has been ex¡llalned earller ln thls chapter that

lf a < c { b then the exact value of c ls lrrelevant to

the payoff obtplnecl. This |s conflrmed. by the fact that

o d.oes not appear ln egs. ?6 and 28.

It is now posslble to soLve this game ana1ytlcaL1y

tn the followlng weJrr Flrst it is aggumed' that

f = o bg).

The cönrectnees aflt conslstenoy of thls aesnmptlon wllI be

shown ln t¡.1¡.2.

Thus, puttlng f=O lnto êÇe. 26 and' 28, gives

P1 = d(1+a-b) ea(1-e)

+ 2At(e-ô) ( t-e-a)+(e-u) (n-¿+ze-e) I ( ¡o) \

Pz = -0(1+a-b) +2a(1-e)

-2e[(e-tl) (t-e-a)+(e-b) (u-¿+ee-z) I (lr¡

Evaluate the followlng partial d.erivatives

uslng €eci. 3O and l1

#f = d-2(r-e)-ze(e-d)

# = -d.+2e(-2þ+n-e+2)

F = -i-a+þ-2Q(a+þ-1)

P = -2a-2e(2e-a-b-1)

and. solving the equatlons

97.

. òP. òP"

--a---^ãã_=" t òb

Ès=., - ?P'=oãF=" ' òe

using Gausslan ellmlnation (aesumir¡g that q. is stloh tfrat

d.iv1s1on by zero iloes not occur) it ls found. thatVão-u4--C1

, b=

z-ct 1z-zø): 1+2Q

where

e= ú + (t-e)z

^ - 2( 1-l+a)v1 1+2e,

Q2=-4Q- z( za-t\2Q+1

ca=4Q- c12

Cr Ct-Cz4q

d ,O=C7

c4 = 2-2q #i+C. -Cøcõ = -â.- - Zq

c"=Ln-Z-i t*#]c7 = îæ4J

and. f or q = .287 tt can be shown that

98'

a=O.O298 b=0'.6418

d.=0.6127 e=O.8106.'glr¡cer âs explalnect ear]1er, the exact value of c ls

lnmaterlal 1f a<c(brLet c=b-0.6418.

. .Llso, by eq.292 f=or thus the complete solution ls

a-O.O298 þ=0.6418 c=0'64t8

c[ = 0.6127 ê = 0.8106 f = o.o0oo

a¡.ril tllls 1s the same as the nr¡¡nerical solution given in

table 11 for Q = 0.287. The nu¡1erical solutions glven

in table 11 fon other values of q may be checked- ln the

SAme Wâ}rr

lr.lr.2 ShcnrlnÊ that the solution found, 1s an e'p

In thls seotion the assumptions mad,e ln eqs.24 an(l

29 rrl11 be val1d.ated., a¡d then the solution w111 be shown

to eatlefy Nashr s crLterj.on.

tr'irst-, eq.'24 follows from eq.32'

It w111 now be shorm that eq.zg (f=O) satlsfies

the Nash conclltlon a¡rd. is thereforê correGtr

First d.lfferentiating eq.?$ partlally ÌYith respect

to 1,

(32)

-?3i.òf = -O.41 < O (33)€.0 OfeQrSã

Flq,33 shows that the assumption .that f=O ls correct, fon

the follorving reason. From eq,.33 player 2 can onLy in-

crease hle payoff Pp by maklng f snaller (if the other

99,

varlables renaln coneteltt) .u 'F ( O. But eince f=O

anÖ f is constralned. to O < f < 1r player 2 ls forced.

to leave f at o, and. hence e4.29 is Justified; The

remaining derlvatlves satlsfy the Nash conèitlons, since

they are zarot

Thus tJre ê.pr found. analytically satisfles the

cond.itlons for a Naeh srPo

L.L.3 The deoend.ence of ttre ana ie solution on the

aPDroxlnate solutLPlg

The above mettroct of analytlc solution le based.

upon certaln agsuriptions (laten proyed. correct) mad'e ln

eqs.Zh and, 29. Ho'yever, it can þe shown that there can

be approxlmately 2a different posslþle lnltlal assump-

tlons. As the algebra requlred. to solve each ind-lvldual

case is lengthy it would be clearly impractlcal to attempt

to solve this problem analytically unless the number of

such posslble cases was flrst reilucecl to manageable propor-

tlons. one way of achleving this 1s to have an approxi-

mate numerical solutlon initially.Allthesolutlonsgivenintaþ]e11haveþeen

checked analytically in the manner d.escriþed. ln thls

s€Ct1on¡

100 -

h.5- Valiilatlon of hand. lnprovenent slmulatlon

In this section lt w111 be shown that the approxl-

nate É1mu1ati orÍ of harid. lnproVenerlt in tJre 2-PG is, for

the purposes of this stucl¡r, ind.istlnguishable fron real

hanil improvement.

Define the z-Pg{ to be a game, ld.entical to the

z-PG 1n every respect, exCept that exact hanit improvernent

(as w111 be ctefineiL below), rather than approxirnate hand'

lmprovement, w111 þe used., and. thUs the 2-PGS is equiva-

lent to phase I of real poker. Í'his section will show

that strategies optimal for the 2-PG are also optÍmal for

the z-Pe¡+.

The functlon f¡(h) was d.eflnect in chapter 2,

a¡rd. d.eterrnines the winning probability , x = fb (h) t assoc-

iated. vfith ar¡y poker hanit h. Define fir(x) to be the

inverse of the ftrnction f¡(h), where h = f;t(x) d-eter-

mines the poker hancl h associated. v¡ith any given wlrrning

probablllty x. Consequently, h = fËt(*) relates arLy

given xr O < x < 1, to some poker hancl h'

Suppose a player holcts hand' x 1n the z-PClÜ 'Ihen hand lmprovement ln the z-PG{ 1s d.eflned. 1n the foIl-

owing wâ)re Flrst calculate hr - f6t(*). Next improve

hx by cllscard.ing the appropriate nrrmber of carcls (see

chapten 2) ancL replacing them wit¡ new card.s (ta¡ing lnto

account card.s alread.y held) to give an inproved' ha¡rcL hl '

r01.

Now it w1.11 be d.emons.trated. that .the payoff

functlons for the z-PW are ind.istingUlshable from the

corresponcLing payoff functi ons for the z-Pe'

Proceed.lng as ln the 2-PG, d'eflne

xi;¿*(*rv) = the expected. winnings of player '1 holctlng

hand. xr conpeting against player 2 holùing

hand.yin2-PG*wherebothplayersimprovetheir tranils in the manner d.escribed- above, and'

player 1 stand-s to elther gain vv units or I'

units ilepending on whether he hold's the

winning ha¡rd or not, (:4 )

and.e*aå,rG v rl

q2( *, y) dxdyx (ts)

It follows from eq..34 arLd 35 that a monte carlo nethod' nay

be employed- to d-efine xqg'zt in the follor'¡1ng way'

V=c

= the expected. winni-ngs of player 1 over a

perlod. of many games of ttre z-Pet where he

stancls to either galn liv unlts, or l'

unlts d.urin¡g any partlcular game (d'epencting

on whether he wins or loseg t'Tre hand), and'

where xr the hand of player 1, 1s sLlch

that a<x(b, arÉ. Yt tTFhand'of

player 2 is such that c < Y ( d' (re)

xeä'¿* /þ

(" )d.c

102.

hypothesls to better than the Y. confld.ence levef.

Now by using eq.36 1t nay be seen that it ls possible to

estlnate xgy'te by sinulating the results of a large

number of games played. uncLer these particular cond'ltlonS¡

Thus xqy'ct may be calculated' to an arþltrary

d.egree of accì,rracy, which is iteterrninect by the nunþer of

games simulated. (max). A Fortra¡r program was wrltten to

do this, and. the resuLts are glven in table 12 below.

Max was J-imltecL to approximately IOrOOO for

reasons of ocofroftfr

It may be seen from the above results tbat¡

(r) Lines (ß) (15) in taþ1e 12 show that when the exact

value of xQä '' {t ls known to þe zero, the approxlnate

value of Xqg'zf approaches the correct value for

increasing values of 94.(t) The expected. orcler of j-naccuracy whlch appears to be

associated' wit'tr the xq|'¿Ç approximatlon fon

max = IO'OOO is approximately O.OO5 (see table 12,

lines 2 anit 6). Furthermore 1t rnay be noted. that

the rnaxinum iLiff erences associated- with XQä'l atñ'

the Xeä,¿t approxlmatlon are of the same order.

If it is h¡rpothesised- that the function Xq¿'r'*

1s a normally clistributed- approxirnation to XQä'1, which

þeco¡nes more accurate As max is increased', then a stand'arcl

statistlcal test (t-test) shows that the d.ata supports thist

t"o* the values consid.erecl t = -O .3g5 where t6(Y') = 2,,45'

103.

TABLE L2

SIMULATION OF z-PG PAYOF'F FUNCTIONS

0.000

0.000

0.000

0.000

0.000

0.815

0.00h

-0.016

0.069

0.068

0.006

o.702

0.039

0.002

-0.023

0.100

0.001

0.006

0.003

0.002

o.8ro

0.000

-0.0L1+

0.071

0.071

0.000

0. T03

0.0U+

0.001

-o.026

0.100

0.002

0.000

0.000

0.000

10,oo0

10,000

10,o0o

10,000

10,000

10,000

I0,000

10,000

10,000

10,000

10,000

10,000

10,000

l+o rooo

8o,ooo

1.0

1.0

o.g

0.8

0.8

1.0

o.7

0.8

0.6

0.9

0.7

1.0

I.0

L.0

1.0

0.1

0.8

0.5

0.5

0.5

o.5

o,2

0.3

0.3

0.5

0.1

0.8

0.5

0.5

0.5

1.0

r.0

0.0

t.0

1.0

1.0

0.9

0.6

0.lt

0.?

0.9

1.0

1.0

1.0

1.0

0.I

0.8

0.0

0.5

0.8

0.5

0.1

0.L

o.2

0.3

0.6

0.8

o.5

0.5

0.5

-2

-2

-2

-2

-2

-2

-2

-1

-l_

-1

-1

-2

-2

-2

l+

2

2

2

2

2

,

5

3

2

I

I2

2

2

(r)

(z)

(3)

(l+)

(:)

(6)

(r)

(8)

(e)

(ro)

(rr)

(rz¡

(rE)

( ru¡

(r¡)

iåPirsexact

*QT''*'(rr rno¿n)xqf 'ß

nax. no.ofganes used.in sim¡14-

tion¿tcba9.'[f'

.104.

Hence from the Above and eq.21 1t follows that

payoff functlons calculated for the 2-PG and Z-PG* can

for the purposes of thts study, consldered ldentlcal.

the optlmal strategy for the 2-PG wltl also be optlnal

the z-PGtn.

the

be

Thus

for

4.6 N -optimal solutlons

sltuatlons often arlse ln real games where one of

the players ls larown to be playlng non-optlmally' It 1s

then posslble to use the LAH algorlthm to flnd straüegles

for the other ptayer, whlch w111 yleld hlm an even better

return. Conslder the example of a 2-PG wlth 9=0.287, wlth

the solutlon (see table tI):-

a b c d e f PaYofftoIPaYofftoII0. 03 0.642 O .642 o.'613 0.811 o .000 0.1940 -0.1940

Note that player II has a payoff of -0.1940 whlch means that,

on the average, he expects to lose 0.1940 unlts per 8ame.

suppose that player I now plays .b=0.500 (lnstead

of b = 0.642). It 1s then possible to solve the game as

before, except now the varlable b ls treated as a constantt

equal to 0.5. The new solutlon ls -:

a b c d e f PaYofftoTPaYofftoII

0.020 0.500 0.500 0.538 0.750 0.000 0'174 -0'u4

Note the changes made in the varlables d and e by player

2 In order to take fuII advantage of playen 1fs bad play and

105,

thereby lncrease hls own expected payoff from -0.194 to

-0.174, that lEr lnstead of loslng 0.194 he now loses 0.I74,

whlch 1s an lmprovement of 0.02.

4.7 Dlscusslon of results

A general dtscusslon of thls analytlc solutlon of

poker ls glven ln chapter 5 1n the context of the 4-penson

verslon of thls game (4-PG). However, partlcular aspeets

of the z-PG whlch lead to stmpllfylng assumptlons used ln

solvlng the 4-PG, are dlscussed below.

(a) BlufflnE þy prayer IThe z-PG, wlth q = 0,287, has the solutlon (glven

ln table tI):-a= 0.03, b=0.64, c =0.64, d=0.61, e= 0.82, f =0.00

F'lgures 6 and 7 sho$¡ that ptayer 1 w111 play wlth a hand x

lf 0 < x < a or b < x < 1. If x 1s such that

0 < x 3 ã: and slnce a = 0.03, thls means that x ls a

weak hand. Hence, to bet wlth such a hand ls to bluff, and

the value of a shows that thls bluff should only be trled'

on the average, t hand ln every 33,

Thls agrees wlth the oplnlons expressed 1n poker

books that thls bluff should. be used extremely sparlngly' 1f

at all .,( 30 ) .

In the complete poker game blufflng ln phase I be-

comes even less lmportant, slnce blufflng can be done fatmore efflclently ln phase 2, when more lnformatlon has be-

come avaflable and unllmlted bettlng 1s posslble.

(b) P1 1ne on after a d.ouble

106.

by the last PIaYer

The solutlon to the z-PG (see (a) abovê) shows

b = c = 0.64. Flgure 7 may be used to lnterpret these

values, and lt ls found that player I w111 always play on

after a double unless hls orlglnal bet was a bIuff.

Thls may be lntult1vely Justlfled because player I

only ever plays wlth a good hand (assumlng he rarely, lf

ever, bluffs, âs dlscussed 1n (a) above). Thus lt 1s

always worth playlng on after a double as lt only costs a

further 2 unlts from player I to have a reasonable chanee

of wlnnlng what w111 now have become a large pot. Thls

same strategy ls recommended ln poker books , (32) '

(c) Blufflns bv the last olaver (plaver 2)

The solutlon to the z-PG (see (a) above) shov¡s t1¡at

e=0.0. From flgure 7 and. the fact that e=0'00 1t follows

that player 2 never bluffs by doubllng wlth a vreak hand.

It w111 become apparent 1n chapter 5 that thls game

becomes too compllcated to solve (ny the methods used here)

1f the rules are extended. to allow 3 and 4 players. In

order to obtaln a solutlon for the 4-person case the rules

must b9 _q11ntlfled.It has been shown here that 1n the 2-PG, players very

rarely bluff (see (a) and (c) above), and a player w111

always contlnue after a double (see (b) above). In addl-

tlon poker books suggest that these same precepts are used

107.

by expenlenced poker plâ,yers (see dlscuesfon above).

Hence, the fol-Iowlng slmpllfylng asSuntptíong were lncorpor-

ated ln the rules of the 3-PG and 4-PG'

Flrst,playersdonotbluff,andsecond'allplayersoontlnue after a double.

108.

CHAPTER 5

SOLUTION OF 3 AND 4 PERSON GAMES (3-PG AND 4-PG)

1 ïnt uctlon

In thls chapter a slmpllfled 3 and 4 person verslon( 3-PG and 4-pO) of the game treated 1n chapter 4, w1tI be

solved. There are, however, several polnts whlch should

flrst be noted.

(1) llhe need for simpllflcatlon of the rules(see 4.2) w111 beeome apparent when the targe

amount of central processor (c.p. ) tlme

requlred to solve the l{-pC ls noted (see

5. 3.3 ) . }'Ilthout thls slmpI1f1cat1on the

amount of e.p. tlme would be many tlmes

greater, and as a consequence, lt would not

be practicable to solve thls problem on

ava1lable computers.

(2) Slnce much of thls work ls a sti.al-ghtforrryard

extenslon of work done ln chapten 4, unnecessary

detalls w111 be omltted. AIso the arguments

used to Justlfy approxlmate hand lmprovement

and the appllcablllty of the phase-l solutlonto the entlre game wlll not be repeated as

they remaln unchanged from chapter 4.

109.5.2 Solutlon of the 3-Pc

5.2.I Rules of play

The rules of the 3-PG and 4-PG are the same as forthe 2-PG wlth the followlng 2 slmpllflcatlons, which have

been discussed 1n 4.7 , and w111 be stated here wlthout

further explanatlon.

(a) The form of the strategy functlons 1s so chosen

that dlrect bluffing 1s not a1lowed. Note,

however, that a subtle form of blufflng 1s

st1l1 posslble (see 5.5.2).(b) When the last player doubles all other players

must look.

5.2,2 Stratesles

In the 3.-PG players Ir2 and 3 recelve hands xry and

z respectlvely, and then must make decislons, according to

the nules of the game, oû the basls of thelr hands and the

actlons taken by other players. Thls 1s most convenlently

represented by flgure B below, whlch glves a flow dlagram

for the 3-PG. It 1s assumed that the 3 players are dealt

hands xry and z. The dlagram llsts aI1 possible game

outcomes and speelfles the probablllty functlon whlch each

player will use to declde whlch actlon to take at any partlc-

ular stage of the game. Each probablllty functlon fi(x)ls deflned 1n terms of varlabtes ai as lndlcated 1n flgure

B.

wt¡ere tî(y) ta theproþabll-l.ty thatPlq¡er 2 Pla¡'e rithhanô y lf pLayer 1hes tllopped.

11(v)

PIJAYER 2PLAYS

I

rl(z) rs the p"oþ.tbet Ésyer 3 p1aüs

1 ft(z)

FIGUNE 8

rl(x)

0

vnere rì(x) ts the prob-abiüty of !14ù'er I play-tng wltb hed :l.

PIJAYER 2DÌOPS

rl( z)

110

PLATEN 1PTJAYS

ttk)

J

lrbere ft(y) ts tbeprobablJ-lty thatplsyer 2 vtlJ- pJ.ay lf.p1€yeÌ 1 lc pLsd¡lng,

f

PIJAYER 1DNOPS

PIJATER 2PLAYSPI,AÍER 2

DBOPS

PI,AÏER 3I{INS BÏDEFAIIIJT

AFTER A DOUBLE ALL PLAYERS

STII,L IN THE GAME MUST PLAY

QN'

tî(z) is the prob.thet plsyer 3 ôoubles

f¡(z) rB the prob.thBt player 3 plEvs

¡i(z)I

rl(z) re the prob.that plaû'er 3 aloúb1i6

f$(z) rs the !rob.thet player 3 p1sys.

rå( z)

f8(z) is the lrob.tbet player 3 aloubleB

rl(z) tå(r)1 I

z

I

z?,

1r1.The follr,rwing points relating to the form of the

probabillty functlons should be noted.

The stnategy functlon for the flrst player in the2-PG was assumed to be of the form f(x), ês descrlbed 1n

the flgure below.

f(x)1

(r )

a x

As has been noted ln 4.7 a non-zero value of a

lndlcates bluffing. As brufflng has been excruded from

the 3-PG, a 1s set to zero, and thus the strategy functlonfor the flrst player 1n the 3-pG, frr(x), takes the form

shown ln the flgure below.

rf(x)

(z)

x

1

a

1r2.5.2. ? Eyaluati on o{ p,avoff functions.

Before the payoff function is evaluated (in tfre

same manner. as for the 2-PG) it is necessary to define

the fol-l-owlng functi on.

Xn'l (*ryr") = the expected. wirurings of playef, 1, (l)q3

hold.ing hand x, competing against

players 2 and. J hold.ing hands y

and. z respectively, in the 3-PG,

wher.e player 1 wins Ìu if he beats

his opponents, or L if he loses.

Now, proceed.lng as before, it is possible toexpress this ñrnction interms of the functions

.Xn ,l (*ry) and. Xw ,t (*rg rz) where2s

xw 't (*ry) = (4)ç ìJy i.x> yL ¿. ;*x < v2

and.

*l't (*,Y ,' )r> zx< zor

x> yx<y)=[ii

l.[-]

ifif

and.

] [-; '' (*,v)]

ß)

(6)

By combining eqs. 3r4 arfl 5 ib may be seen that

xäå' (*,v,u) probability that hand. ximproves while hand.s y and z d.o not

.I

.I

probabllity that at least 1 of hand.sy or z improve while hand. x d.oes not

probability that hand.s x and. y im-prove while hand. z d.oes not.

].[,]

.[Tili:?å';li..JTlä"]uå"ä""î3ri] . xi, ¿ (x, z)

113.

xn '2 (xry rr)3

.tprobabillty that either all hand.simprove t et all fai l- to inprove a

-)

)

Now:from the d.efinitions given by eq"1, chapten l+,

1t follows that¡-probability of improving hand. - q. Q)probability of failing to improve a hand. = 1-q (S)

and. combining eqs. 6, 7 and. I it may be seen that

xJ"" (x,y,r) - w[e(t-q¡23 + ¿Í.(r-q)t1-(1-q),]i

* x: t (*,y) [q" (t-q) j * x: ,t (x,z) [q,(1-q) ]

* x: '2 (x,r,z) [q."+( t-q)" I

which simplifles to

xi¿t (*,y,r) = wq.(t-q.)' + ¿(1-q) [1-(1-q.)'] (g)

+ [q'( t-q) ] l*i't (*,y) * x: ,¿ (*,") I

+ [q." + (r-q.)"J.*i ,¿(x,y,z).

Since from ttris polnt, the triple integrals of *"nlt (*ryrz)and. Xl,t (xryrz) arise repeated.ly in the text, it is con-

venient to d.eflne, ("" 1n the 2-PG),

xa: 'G::; l:) = f' f' / o'

*iå'(*,v ,z)dxdvaz (ro¡

X-ãt V=àz Z-ds

and.

114 .

wq.( 1 -q)2+¿ ( r-q.) . [1-( 1-q)']

X!! tt (bt b= bu\ =t \*t a2

^u) f" f" f'*:''(*,v,2)oxavazX-âr Y=àz Z-Ag

then by substituting ee.9 lnto eq.10, anil using the above

equation, it can be sh,own that

*a: ,Gl f ñ (¡'-*,)JLr=r ))â3â.2

þ, bP bs\\u, az

^t )

where the functions Yw ,x (bt ¡,\' \t" ut)

bP bg

q.3 + (1-e)"

I

t

+ q.a(1-q).[,o"-,")xX'rGi 3:) . (b,-^,)x:, þ, bg

\tt âg )l

+ .Xw ,l3

(tr)

and *;',Gî k 3:)

have been evaluated. and. are given 1n append.ix. A.

Fnon this point the calculation follows the pattern estab-

lished. in chapter 4. Ffrst, all possible plays, theirprobabilities of occurring, and. correspording payoffs, are

calculated., and. t}¡ese are given in table 13.

Next, proceed.ing as in the z-PG, and making use of

table 13 and. eq.11, the payoff functions for each player

may be evaluated.. Si-nce the method. of caLculati on is the

same as for the 2-PG, only the final resul-t will be given

here.

115

PIav

TABLE 13

STJMMARY OF PLAY FOR THE 3:!q

Probabi]-itv PaLoffs

-1

xln'2(z,y)

xl;-b(,,r)

-1

xf,'-2(z,x)

tå'-u( "*)

-1

*å;-'f z,x,Y)

*åi*( z,x,r)

1

x';r''G ")x[;-u (r, r)

0

0

0

xl; -'(r,")

x[;-'(v,*,")

xfr'-a(v,x,z)

+1

xT[ "G,")

ya '-r(x.z)"Qz

xf,'-2 (x,v)

xl; -2 ( x,r, z )

*å;-*( x,r,z)

0

0

0

DrP2D3

DrP2P 3

otp'o8

P rD2D 3

PrD2pg

plo'4

P rP2D3

P1P2P3

pt"'4

lr-rï(") lrl(y) tr-r1( z)-t?z(z)l

[r-rì ( x) ] ti (y) tï ( ")

tr-tÌ(x) lr?(v)rl(z)

rl ( x) tr-rå (y) I tr-r3 ( ,) -ri ( ") l

tl(x) tr-rã(y)l13(z)

rl(x) tr-râ(v)lril(z)

rl ( ")

tâ (y) tr-r3 ( z) -rt( z) l

rl(")rã(y)18(")

rl (¡c)rä (y) rt ( z)

I2

3

L

,

6

T

I

9

TII - rïr

116.

Define the jolnt etrategy vector+

a - latra|ra|ra|'ra}ra$ralra!ra!]' (lz)^l

anit let Bå(d be the correspond.ing payoff to player i,for i=1 e2 and. J¿

Then, it may be shown thati

Bâ (e) = a?a8?-a1)+a3xeB '-. Gl 3ã)

+ alxa,A,-'(^l "l )

. agxQB ,-, (^l "å )

+ xQÉ ,-"(^! "L 3Ë) . xea, -,

Gi )1

aflI

^34g

p3(e) = at(r-a?)a? + atxo3,-,(Lrå?) . alxaå,--(^l J)

+ asxoa'-'(,â l*).*'-'Gâ .l åË).æ ,--Gå ^1 .å)

PB (e) - al(t -u?) a? + atxqï'- " (?q "i)+ atxoå '-'(; "i) - U-^T) af;a!+af;Næ,- "(äå "i)+ øAxq,t ,-. ("1

"1 )

+ x8å-,(3ã

(t-a1)(t-a3)ag

1

a!1

al )Å "¿) . x.s'-'Gä

117.

Constralnts

By virtue of the f\rnctJ.on deflnitions (see figureB) 1t follows that:

o<aJ <

and.

al<

Knowleclge of these d.ata perrnits solution of the

l-PC by the lÆI algorithm.q-D Sn'l:r t i on of f.he 3-Þêlra

The soLutlon of the ]-pe' for e = 0.287, lspreeented. belon¡.

PlaVer Expected. VarlaþIesPavoff

.,1

2

3

o.1186

o.1g3g

-o.3225

a{ = 0,715

al - o.61i

a! - 0.620

a! - o,BJo

afi = 0.800

afl - 0.815

a8 - 0.825

a$ - o.73o

a8 - 0.885

This eolution will be d.iscussed. 1n 5,5.1+.

the LÆI algorithm took approximately ly'+ second.g of

Central Procesgor (C.p.) tine to conpute ttrls solution.6.1 Solution of tlp h-PG

5.5.1 Evaluatlon of payoff, f\rnctions

The rules of the h-PG, apart from the addition of

1 extra player, are exactly the same as for the 3-PG.

I1B.

Also the method. of evaluating the payoff fr¡¡rctions remalns

unchanged., except that the foIlowlng preliminary result isfirst required..

Before eval-uating the payoff functions for the

z-PG , th e functl on XQg , t (?'\u"

ated.¿ Sim1lar1y the f'unctio

required. before the f-PG payoff fr¡nctions could, be calcula-

ted.. Now, following this pattern, the function

xgl ,¿

w111 be d.efined. and. eval-uated. as a prerequisj-te to calcul-a-

ting the 4-PG payoff functiofrʡ

tr'irst, d.efine

xw,t (*ry rzrw) = the e4pected. winnings of (ll)^qr,

player 1 in the 4-PC

hold.ing hand. 21, competing

against players 2r3 and. 4 who

are hold ing hards V tz anit ït¡

respectlvely, where all players

are given an opportunltY of

lmproving thelr hand. with Prob-

abllity eo

b.\^r)

þ" ba bg

\"r à2 âs

Now, as for the J-PG,

terms of the functionsxw 'tr (xrY rzrw)q1

may be expressed. in

119.

t,

*î" (*ry, zrw) = [;

xw '¿ (*ry) =2

xw 't' (*ry ru) =3

By combining eqsL 13 to 16,

, x > yt x à z, x Þ ïr;x<y or x<z or x<nr

[;x> yIt<y

xÞ zx< z

and.or

! x> v;x<y[;

a w

( 1l+)

(tt)

(tø)

(17)t*î;t (*,v, z,w) = [

].[']

probability that hand. x inproveswhlle harda yrzrw d.o not

*[nr9,tabi1ity that hand. x d.oes not improve'[-and at least one of the hands vtztw d.oes

+

+

+

["rr hands lmprove except hand.

"]j[-: ,t(*,r,ù)

["u hands in¡rrove except hand "].[*;,¿(*,y,*)]

["rr hand.s improve except hand. *].[r" ,t (x,u,ù)

+ only hand.s x and. y

only hand.s x ard ïÍ lmprove

inprove ].[-:

,, (*,v) ]

+ only h,arrd.s x and z improve

.I]'[-: "

(*,ù)

]'[-; '' (*,")]

all hand.s lmprove or all fa11to funprove concurrently ].[-: '2 (*,v,",ú)+

L20.

From eqs. 7 *nd. 8r eg.1/ reduces to

,i;t (*,y, z,w) = nq.(t-q.)s + ¿|(t-q)t1-(1-q.)"1J (1s)

+ q."(1-q.). [*],, (x,z,vt) " X: ,t (x,vr,u¡)

" X: ,t (*,y,r)!

+ q.2(1-e)'IXI ,t(xry) + Xl ,r (x,z) + X*,¿(xrw) J2 '-2 .-2

+ [ e¿ + ( t -q) t l*i,t (*,y ,z ,w) .

NoÌrr, as before, d.eflne

xa; 'Ç:H l: ï) = l"' f" f" f'*:;t(*,y,z,n)òcô¡òcdr',x-a1 Y=tz Z=a.g ïV'=44

and

Xw ,2I

x-41 Y=ãz z=ag lll=â¿,

thus using eqs,1I, 18, 19 and, ZO,

be shown that

x8

+(br-a")x, 't'3

*i, t (*, y, z ,w)ù ü dz dr

(zo)

and- integrating, it can

(1e)

(bt bz bs b.\ = P'\ut ã2 â3

^*) lI

[, !, (b' -"r,

]" I wq.( 1-q )' + a (1 -q.) I 1 - ( r -d " J ]

n,tl'bt bB bs b¿\. \.r d2 âs

^o)

þ, bB b¿\\.1 à2

^* )+ qs ( 1-d[ (b. -u. )*: ,, Ë k :: ]

(b" -"" )*; , ,

þ, bs b4

\ut âs ã1 )l(br-^r) (b"-*" )xn , z +(bz-az) (b¿ -ar)xn 't

b4

)3

)b+q.3 (r-e)"

2 à¿ 2 8s

l-2T.

+ q.*+ ( t -q.)' Xw 'l

(b"-"") (¡* -^r)*i't )l(bt bz b3 ¡*\\., à2 â3 u'/

ßt bz

\"t ã2

(zt)L

Figure ! belou defines the game flow d.iagnam and.

the strategy firnctions for the \-PG in the sarne way that

figure I ttras used. to ilefine t}re same aspects of the 3-PG"

fn this d.iagram all posslþ1e game outcomes, and. theirprobabilities of occurrence, are presented-. ft isassumed. that players 1r2r3 and. 4 are dealt hand.s xtvtzand. Ìv respectively, and that each playerr s actions are

governecl by probability ftrnctions of th.e form f](x)which are d.efined. in terms of variaþles aj. Defi-ne, a,

the jolnt strategy vector for the 4-PG as follows:

? = laï,a?ra?,a?,aB,a8 ,a? rai.rab,abrat,aå , aå ,al ,ab ,4 rat o ,at.t rat z rats , ut

"fTIhe next step is to summarize all posslble p1ays,

and. to speclfy the probability of each play occurring and

the correspond.ing payoffs. This is done in table 14

which follows ttre sare pattern as table 13 for the 3-PG,

with the follon'ing exception.

It is clear that if player 1 d.eclines to play 1n

the 4-PG then the game becomes equivalent to a 3-PG

(see figure 10). Thle fact may be used. to reduce the

amount of computation necessary to compute the payoff

G¡,fr FIOW DIAGRA¡.I ¡þR

rHE !-PC Aì{D DE¡IüTTION

OF ST¡ÀTEGT FUICTIONS

fî(z):the prob.thatllq¡¡e} 3 eDters

ft(v):prevfú({):double

FICUBE q

rï(v) ¡tn" prob.that?lqyer 2 entelB

fl(z):tr¡e prob.thatplryer 3 entèr€

fi(x):the ?rob.thstplsyel I ebte¡e

t-22

rå(y) ¡tle prob.thêtplryer 2 eDter6

v

f!(r),fir. prob.thetp1ryè¡ 3 enters

PI,AYER 3

PI,AYg

PLAYDR 1

DROPS

PTAYEN 1

PLAYS

1

PLAYER 2

DNOPS

vPLAYER 2

PUTYS

PIJIYER 2

DROPS

PLAYEN 2

PIAYS

PLAYBN 3

PUTYS

PL$EN 3

DNOPS

z

PLAYER 3

P¡AYS

PTAYER 3

DNOPS

PúÀTEN 3

DROPS

o

fi(z):tbe Þrob.thêt!1Áù¡er 3 eßterg

PLTqYER 3

PLAYS

1

Fmllmpsl

rT(w):Þr€y tf!(tr):doubr.e

IIII

f!(v):ple¡få(v):doulre

få(v) rÉâyf3(*):dou¡te

13(v):p1avfTo (f) ¡¿oub1e

rlr(v):pIrvfTz(v) ¡doutle

fT¡(w):prqvf1¡(t):double

tgglg Íhe ùove represlDtêtlonof the pair of füctionB fl(v)md ri(s) is choseh to conserve6pæê, The @elng is thet plqyer4 plrys lf sT<y<sä ed ¿touhlesIf eå<s<l,

TASLE 1¡+

SIJMMARY OF PLAY ¡OR TIIE \-PC

L23

+ pi,pl,p3 are the payoffs for the 3-PG

P3'-1

x!'- 'z(',x)

x[, -q (v,x)

-1

y\ ' 2(v,x,z\q3I -4lxqj \w,x,z/

-1

xl;''z(''x'vtxl;-'(* '*'r )

xl; -'(*,*,v , ")xlf '- u(",*,v,,)

32P

0

0

xi;-'z(",x)

xf,'-'(" ,*,t)

x['-f(z'x,w)

0

+

0

xl; -'z ( " ,x ,Y)

x['-'z(z,x,v,v)

xf,''-u(r,*,v,w)

0

0

0

0

xl; -'(v,*)

xl; -'z(v 'x 'w)

x[;-a(v,x,w)

xl;-'z(r,*,2)

x[;-'(r,*','*)xlf '-h(v'"','')

Pi

0

+1

xl;-'(" '*)xl;- b( x,v)

x3n' -2 (x,z)

x[;-'z(x,z 'v)x['-h(x,",v)

xl;-'z(x'Y)

xl,'r (x,v,w)

x[.'-r( *,v,w)

xl;-'z( x,v , z)

x[, -'(*,v,",")

xlf ' -'(",v,,,*)

DI

p lDaDeDrr

P 1D2D3Ph

P lD2D3D{

PlD2P3Dr

P lD2P 3P I

r 1o21 3ol

P lP2DsDq

P lP2D 3P q

r rr2o3ol

P lP2P3Da

p lp 2p 3prr

e rr2e 3ol

lr-rl(*) lrl (x) tt-r3(¡') I Ir-rt(,) ] [r-rf(rr)-rå(w) ]

tl (x) [r-râ (v) ] ir-r! (v) lr](w)

rl ( x) [r-r3 (v) ] tr-rå(v) lrå(*)

rl(x)tr-rå(v) lr3(v) Irå(") -rls(w) l

rl (x) tr-ri(y) lr3(v)rå(")

ri(x) [1-rã(v) ]13(v)rlo (v)

rl ( x) tâ(v) [r-rf,( z) ] [r-rl r (w) -rÏz(w) ]

rl (x)râ(v) [r-ri( z) ]rl r(w)

rì(x)rã(v) [r-ri(,) ]rl,(")rl (x) rå(y)ri (z) Ir-rl ¡ (") -rT,*(") l

rì ( x) 13 (y) r,1( ,) rl. (")

rl( *) râ(v) ri (,) rl,, (v)

I2

3

l+

5

6

7

I

9

t0

t1

L2

L3

I II ]II TVPLAY PBOBABILITY

724.

functions fon the lr-PG, by making use of the ,-PG

payoff functlons which have alreacl¡r been calculated.¡ (see

J.2,J).let pt (g) be the correspond.ing payoff to player 1

for jolnt strategy g. Then, using table 1l+ and figune 9,

the fr¡ncti ons pt (g) may be calculated., ln the ueua1. ïvâyr

and. lt 1s found. that:

/1 aþ

\'t aiPi(e) = (t-a1¡alalal + afiaflxqf;,*z

+ af a!xs! ,-. ("1 .¿ ) - a?,a!x.ql,- " (^l

1

)^34g

+ aãxaå,-"("1 "å åå") . aaxaa'-.("1,å

"1")

a?atrxQ! ,-'Gi "L

+

+ a?xaå 't'(.1 ;i:) . afxqs '-+ ("11

421

afr

)

.T

a!

1

^3At

'la3,

+)

1

al1

?

)

+ xeg ,-,(^I"l :î:) * x81''-,'(

af"Xqg ' - z

+ afxqg '-z

1

^2az

a*

"l1

27-

+

a! at.

Iafl

( al

1

al )L

1

al

På(C) = 4P1(af ,al ,af,a!,ah,t,af ,a!,al)

1

a!( ){

al

1

afi

(

(

)

,l1

1

+ aP aÍ, rx88 '- 2 al al

"?xQg '-'

+ af "Xq[ '-z

11

a!1

a!1

af;

atÞ} (a?, a?, a2, aL, al, a8, at, aE, aå )

1

al

I^¿4g

+

aïat r(t-^t) (t-"7)

( "l) . *æ '-'(

"1) . afixaå,-"("å

1

aZala!^3A-t

Iaf

r25.

L +3

a!a!

1^1.4Lt

)

+ afla$Xgsrt"z

xQÍ, ,-, (^2

alxqg,--("å

) * xee,-"(";

1

a!"1. )

På (e)

( ^9a3II

11

a!

Ial

ii').+

"1" )'1+ ataXQã,-2 al at afl

1

al

(+ XQÍ2 '-1 Ia!

1

^2a2 :) .

1

^24Z

Pt (e) atPg (a7, a?, asz, aLt, ãL, ab, at, aE, ab)

( t-"1) a?a8a1

af;aflxqg ,- " (?+1

a! + afla$xq1 ' - +(

1

al

a?ab(t-aL) (r-.3)

+ azxqb'- 2

+

/ato 1 1\\"9 a!

"8 )

1

al

+ afxqg , - +

a?xQ8 ,- À 1L1-2("

ai" ( i-aï) (t-"7) ( 1 -a? )

11al,o al(

1

^343 )

alxaá ,-'(ii:

+ xef ,-,(Zii

1

^24Z )+ .å) -1

a!

1

^sCL,

1

a!1

a!1

^2e2 (1

)1

^8CL'+ XQâ2 t-L

^44!LI

af;

L26.

5.3.2 Special _method. tg spe_ed. lhe solutlon of the lr-PG

It is not practlcal to apply the L¡AlI algorithm

d.irectly to the task of solving the [-PG for the following

reason. Experlmental evld.ence showed- that:(") Each fìrnctlon evaluation for the 4-PG took

approxlnately

(¡) 2016 ftrnction evaluatlons ïyere required. per

cycle of the LAII algorithm.

Hence, tlme per cycle was approximately 50 secs. Since

many cycles (see chapter 3) are need.ecl. to find. the êrpr

it would. be ad.vantageous 1f some way could. be f ound. torecLuce the cycle time.

In fact the number of function cal-ls required per

cycle may be red-uceil in the following way. Consi der the

tree d-iagram given in figure 10. The section of the

tree d.iagrarn enclosed. in the dashed rectangle is a subset

of the [-PG which arises when player 1 decl-ines to play"

This subset is exactly equivalent to the 5-PG. Thus, the

solutlon alread-y founil for the l-PG may be used. in t'his

section of ttre [-PG. (ttote that tiris fact has already

been used. in table 14). Thls then red.uces the number of

unknown variables in the [-PG, and, as a consequence,

experiments show that ttre number of fìrnction ca1ls per

cycle of the LÆI algorithn is reiluced. to 666. The

evaluatlon time per function remains at approxinately #

fr ".". of cnpo time,

FTGURE 10

EQUÏVALEN CE BETI,\IEEN THE 3-PG AND

THE l+-PG I,üHEN PLAYER 1 DECLINES TO PLAY

PLAYER 1

DROPS

PLAYER 2

PLAYS DROPS

127

PLAYS

rI

PLAYER 3

DRoPS A ,l,O,,/\

PLAYER 4 PLAYER 4

DROPS DROPSPLAYS PLAYSDOUBLES DOUBLES

PLAYER 3

DRoPS A ,.O,,/\

PLAYER 4 PLAYER 4

DROPS DROPSPLAYS PLAYSDOUBLES DOUBLES

2

PLAYS

PLA

DROPS PLAYS

PLAYER 4PL YER

DROPS

DROPSPLAYSDOUBLES

DROPSPLAYSDOUBLES

DROPSPLAYSDOUBLES

DROPSPLAYSDOUBLES

128.

of a second., arxe thus tl.e overall erecution time per

cyc1e is red.uced- to th secs. Thls is a factor of 3 times

faster than þefore.

C onst raints

An examination of figure 9 shows that:-o<

and-

al<

a$<

6.3.3 Solution of lr-PG

With the above d.ata it ls posslble to solve the

l+-PC (witfr çl = 0.287) by using the LÆI algorlth¡n. the

minimun step size used. by the algorithm was O.OOO45 anil.

the executlon time was appro:rlmately J20 second.s of c.pr

tlme. The soJution is glven in table 1þ below.

Checkl the PG and- PGa

Both the f-PG and. 4-PG payoff firnctions were

tested. ln the same ïrâIIr hence d.etails are only glven forone of the above cases. The l-PG case was chosen to

maximize simpliclty of presentation.

Two tests Ìvere carried. ogt.

(") CheclçL¡rg.that the sun of .

Since the ]-PG is a zero-Êum gane (see chapter 3),

it follows that for argr given ioint strateglr vecton a

(see eq.12)

L29.

TABI¡E 16

8OÍ,UTION TO TI# l+-Pe

Plaver EJrPecteclPgvoff

o.113

o.129

o.169

-o.41 1

Variables

-

1

2

t4

tt=O.79O

af =O.715

a|=o.615

af=0.6?0

at=O.825

aå=O.83O

af e=0. 840

afl=O.815

aB--O. BO0

a!=0.815

a!=O.88!

afe=O.90O

afa =O .999

a$=O.B7O a8=O.925

af,=O "73O af =0.830

af=O.800 aå=O.88O

atr=0.8J0 ata=O.92O

130,

P?(e)+PÊ(e)*P!(g)=o

A rand-om numþer generator was used to generate an

acceptable strategy vector pt and. ttris parameter was

substltuted. lnto the I.H¡S. of eq.22 to check its vafidity'This process was repeated. a 1ârge number of times, and. 1n

each case eq.22 was satisfled..(¡) Use of simul-ation as an alternative nethod. of

calculating the payoff function.The payoff functlons for the 3-Pe may be approx-

imated. by using slmulation, in the manner to be d.escrlbed.

below. Given some strategy variable, 3, many games are

simulated., ar¡d. the average ex¡ncted. payoff accruing to

each playerr âs a result of this particular value of ltis then caIcuIated.. Each ind.ivldual game is einulated.

in the folJ.owlng u¡ay. A rand..om numþer generator (see

append.lx B) 1s used. to deal hands xty¡ and. z to players

1 12 and 5 respectlvely. Next, the joint strategy vector,

lt pred.icts the course of events 1n this game accord.ing

to figure 8. For example, if x < a! then player I

d.rops. Next, if y > a?, eâÍr player 2 wlLl PlaY'

rryhile player 3 w111 d.rop 1f z <

Finally, the rand.om number generator ls used- to

effect hand. improvement accord.lng to eq-.1 in chapter 4"

this is cl.one by generating a rand.om number, r, O <

and. allowing the hand x to improve to X - x+2 if

(zz)

IJ]-.

O < r < q. (q- = o"28-l). If çL < r <'1 then the hand'

does not lmprove, ancl X=X. After this has been d.one

for all hands, the eventual ïuinner is decid.ed. in the usual

l,t¡â9.

Thus, for any glven 3 the payoff functlons may be

calculated ln 2 ways. First, by using the analytlc resultsglven 1n 5.2.3 and secondly, uslng slmulatlon to glve an

approxlmate answer. Thls experlment was carrled out forsevenal dlfferent values of ?, and the 2 sets of results

agreed each t1me. The detalls of the least favourable such

result w111 now be glven.

Let

[0.715, 0.615, 0.800, 0.645, 0.815, 0.730, 0.830,

0.825, O.BB5lr

The 3 payoff functlons for thls value of ? were ealculat-

ed analyt1ca11y. Next, the same payoff functlons were

approxlmated by slmulatlng 100 games. Thls procedure was

repeated several tlnes, each tlme uslng progresslvely hlgh-

er numbers of games 1n the slmulatlon. Then the results

obtalned were analyzed ln the followlng way. Conslder

the payoffs obtalned for the 3 players by slmulating 100

games. The relatlve error between each of the 3 playerts

payoffs, and the correspondlng payoffs calculated from the

payoff functlons, may be computed, and the maxlmum relatlveerror of the 3 determlned. Thls turns out to be 647l.

a

r32,

Thls procedure is repeated for the payoffs obtalned by

slmulatlng 101000, 30,000 and 1001000 games, and the results

are graphlcally represented by flgure 11. T'lgur'e 11 shows

that the payoffs obtalned from slmulatlon approach the

analytlc payoffs, as the number of games slmulated lncreas-

es. Slnce the slmulatlon of 100r000 games took approxl-

mately 30 seconds of c.p. tlme, and eonvergence appeared

slow, the experlment was termlnated at thls polnt. ClearIV¡

s1mu1atlon, as a method of obtalnlng payoff functlons, 1s

slow and not sufflclently accurate for the purposes of thlsstudy. It 1s, neverthel,ess, a useful standard agalnst

whlch the analytlcally evaluated payoff functlons may be

tested.

The same method was used to check the payoff

functlons for the 4-PC.

5.5 Dlscusslon of results

5.5.I Comparlson between the 2-PG and 3-PC

Deflne the game 2-PGl{ to be the 3-PG where the

flrst player has decllned to p1ay. Thus the 2-PGlÊ lsequivalent to the 2-PG (conpare wlth the 4-PG reduclng to

the 3-PG when player 1 does not play, âs descrlbed 1n

flgure 10). In thls sectlon the differences between the

strategles adopted by player I in the 2-PG and 2-PGlt wlllbe dlscussed.

F']GURE 11

COMPA RISON BETI^IEEN ANALYTI C PAYOFF FUNCTÏONS

-PG D RESULTS TAINED FRO

133

TfON

7o

6o6W' ercor

error7% error

2% error

100 .]000 lQr000 100r000

number of games slmulated

50naximumrelatlve h0/o etroÏ.over the 303 p}ayers

20

10

134.

In 4.3 tt was shown that, ln the case of the

z-PG, player I w111 play, only 1f hls hand x 1s such that

0 < x < 0.03 or 0.64 <

However, because of dlffenences ln the assumed form of the

strategy functlons for the 2-PGlß (see dlscusslon 5.2.2),the solutlon to the z-PGt¡ shows that player 1 w111 only play

1f hls hand. x ls such that x > 0.61 (see 5.2.4). Itw1ll be shown that, for practlcal purposes, these two

strategles are ldentlcal.Flrst, 1n the case of a hand x, x > 0.64, both

strategles requlne the player to enter the game and are

thus ldentlcal. Now conslder what happens when

0.61 < x < 0.64. The z-PGlÊ strategy requlres the player

to enter, wh11e the 2-PG strategy does not.

However, Íf player 1 does enter, and he 1s looked

ât, he can only posslbly wln 1f the second player¡s hand y

ls ln the range 0.62 < y < 0.64, slnce the second player

w111 only play on a hand y >

above, lf player I has a hand x, 0.61 <

player 2 has a hand V, 0.62 < y < 0.64, then player tshould wln half the tlme. That 1s, h1s chances of wlnnlng

1n such a sltuatlon are

þ x (0.64-0.6r) x (0.64-0.62) = 0.0003.

Hence, for practlcal purposes, player I may lgnore the

t35.

posslblllty of wlnnlng ln this poel_tlon.

ConvenselÍ, lf x ls such that 0 < x < O.O3

then the 2-PG strategy requlres a player to enter, wþ11e

the 2-PGlt stnategy does not. uslng the same arguments as

above lt follows that ln thls second case the practlcalresuLt again 1s that player I may expect to lose 1f player2 looks,

Thusr lt has been demonstrated that the practlcaleffects of player I elther playlng on hands x,

0.61 < x < 0.64, or of playlng on hands x, O <

are the same. slnce the frequencles of both occurrences

are equal (0.64-0.6f = 0.03), lt follows that the two

apparently dlsslmlrar strategles have a very simllar prac-

t1cal effect.5,5.2 Pnactlcal ïnte rnre t 2 tlon of the solutlon to the 4-pq

Thls sectlon w111 dlscuss the practical interpreta-tlon of the solutlon to the 4-pC.

Chapter I presented a survey of the llterature on

poker-Ilke games. It was then polnted out that thlssearch falled to flnd any games tlrrat could be dlrectlyrelated to any commonly pl_ayed varlety of poker. fn the

ensulng discusslon it w111 be shown how the solutlonsobtalned here may be used ln thls way.

Before thls can be done two prellmlnary nesults

are requlred.

136.

(f) Relatlne a real poker hand to some hand x ln the 4-PG

Any real poker hand, h¡ mâV be related to a hand

x ln the 4-pC, 1n the foIlowlng way. The work on poker

slmulatlon requlred the deflnltlon (see 2.8) of a functlon

fb(h) whleh gave the probablllty of a hand h beatlng any

other randomly dealt hand. However, any hand x ln the

4-pC, by vlrtue of belng evenly randomly distrlbuted on

(0,1) (see 4.1), 1s numerlcally equal to the probablllty

of beatlng any other randomly dealt hand y. Thus, 1t

follows that any poker hand. h ls equlvalent to a hand

t( = fr(n) m the 4-PG.

The algorlthms developed 1n chapter 2 may be used

to evaluate f¡(h), the probablllty that a poker hand h

has of wlnning. Table 16 glves the values of these prob-

ablllties for varlous types of hands, and, âs a result of

the above dlscusslon, âñ equlvalence 1s thus establlshed

between h, a real poker hand, and x, a hand deal-t 1n

the 4-PG.

(2) Correspondence between a elven hand 1n the ll-PG

and a poker hand

As a consequence of (1) above lt 1s posslble to

provlde an lnterpretatlon of any hand x 1n the 4-pC tn

terms of a real poker hand h. Thls procedure ls best

lLlustrated by an example.

r37.

TABT,E 16

EQUI VALENCE BEÍT,IEEîI IN ITIE AIVD NEAL POKER IÍ.AIVDS

o'50

0. 53

o.57

o.6o

o.6l+

o.67

0. ?1

o.Tl+

o. z8

0.81

o. 85

0. 88

o.92

0.98

OR HIGHER

OR HIGHER

OR HIGÌER

OR HIGIÍER

OR HIGI{ER

OR HIGHER

OR IÍIGHER

OR flIGiER

OR HTGHER

OR HIGÊIER

OR HÏGIIER

OR HIGHER

OR HTGHER

OR'HIGHER

OR BEETtsR

OR BEITER

OR BESTER

OR BETTER

OR BESTER

OR BETTER

OR BETTER

OR BETTER

PAIR 0F 10rs OR BETÍtsR

PAIR 0F Jrs 0R BETIER

PAIR 0F Q's 0R BET1ER

PAIR OF Krs 0R BET$R

PAIR 0F Aces 0R BETIER

TITREE OF A KTND OR BEÎTER

NOÍTIING

2fs

3rs

l+rs

5's

6ts

7ts

Its

9ts

PAIN OF

PAIR OF

PATR OF

PAIR OF

PAIR OF

PATR OF

PAIR OF

PAIR OF

x = ¡u(h)

POKER IIAI\ID. h l+-PCEQUIVALHVT HAND IN

138 .

It was shown 1n table 15 that the optlmal solutlon

to the 4-eC fra¿ al = 0.790. By referrlng to the orlglnalstrategy function deflnltlons (flgure 9) lt may be seen

that thls requlres player 1 to play 1f h1s hand x 1s

such that x >

Table 15 shows that a palr of 10 | s or better 1s

equlvalent to a hand x, x > 0.78, wh1le a palr of Jacks

1s equlvalent to a hand x, x > 0.81. Thus x > 0.79

means that hands h weaker than a pair of 10rs are never

played on, while all hands stronger than a palr of 10ts are

always good enough. However, 1t ls not lmmedlately clear

what actlon should be taken when L.clding exactly a palr of

10rs, and thls problem 1s resclved ln the followlng way.

Flrst.;'two hands both contalnlng one palr of 10rs

may be ordered on the basls of the 3 remalnlng cards (see

table I, chapter 2). Thus x > 0.79 means that only palrs

of 10ts of above a certaln strength (Judged by the 3 non-

palred cards) are sufflclently good to play on. Slnce

the probablllty of obtalnlng better than or equal to one

palr of 10's 1s 0.78 and better than or equal to a palr of

Jacks 1s 0.81, then the probablllty of obtalnlng exactly 1

palr of 10ts ls 0.81-0.78 = 0.03. If now only palrs of

10ts are eonsidered when x >

obtalnlng such a hand 1s 0. 81-0.79 = 0.02. Thus, thlsanalysls shows that x > 0.79 lmplles that only 3# = ?

139.

of al1 palrs of 10Is dealt wlII be sufflclentLy good to play

on

S1nce, ln a real gane of poker, the 3 non-palred

cã.rds th the hand are usually dlscardeil, from a practlcalpoint of vlew, all palrs of 10ts are equlvalent. Thus,

uslng the above two ldeas, Lt ls posslble to lnterpretx > O.7g to mean that a palr of 10's 1s only played o" ?of the time. Thl-s strategy would mean that a player could

make a random cholce on whether or not to play wlth a palrof 10fs, provlded. that, oD the average, hê played Ç of the

J

tlme,

If, as mentloned by von Neumann, (27) " one of the

obJects of blufflng ls to create uncertainty ln the opposlng

players, then, the above apparent randomness (1.e. playing

only two thÍrds of the tlme wlth a palr of tens) fn the

strategy, could be lnterpreted as a form of bluff.Each of the varlables aj may be lnterpreted 1n

the above manner, and the resultant, overall, strategy, lsglven in flgure L2. It should be noted that ln thls flgure,only the probabllltles of playlng wlth the crltlcal hands

are speelfled, as 1t 1s automatically assumed that hands

weaker than thls are never played on, wh1Ie stronger hands

are always played on.

5.5.3 Dlscusslon of solutlon to the 4-pA

Thls sectlon w111 discuss the solutlon to the

I{-PG glven ln table 15. A comparison wlll then be d.rawn

F'IOUIIE ],2 : SOI,I'IION fO TI{E 4-PO EXPRESSED IN TERMS'OF POKER HÀNDs 1q0

In the dlagrm below abat€nentE of. the forn should òe lnterpreted ln bhe followln8 xay'

A plåyer shoulil never play lf hls hsnd 16 , ¡¡id always play lf hto !¡and 1s Etrôngsrthù e pelr of têns. Howevê¡, 1f he haB exactly ¿ ¡rê1r of tenE then ha shoulal qnl'y play Flth proÞabt¡.lty 0.7.

PLAYER I

DROPS

PLAYER 2 PLAYER 2

PAIR EfcHTs, PROB-o.8 PAIR JÀCKS, PR0B-0,8

DROPSPLAYS DROPS PLAYS

g¡lE¡l PLAYER 3 PI,AYER 3 PLAYER 3

PAIR FMS, PR0B=0. 7

DFOPS PLAYS

PIJAYER II

PÁIÌ TENS, !ROB-o,3

DROPS PLAYS

PLAYER 4

PLAYER {

PAIR JACKS ,PR0B-0.5DOUBLES KIN0S,PROB=0. I

PAIR qUEENSt PRoBro,3 PAIR ACES, PRoB-1.0

DROPS PLAYS

PLAYER 4

PAIRDOUÉLES

TENSKINOS

¡PRoB¡0,PnOB.1

30

DROPS.PLAYS

PLAYER II

PITAYER 4

PÂIR JACKS ,PRoB-O,5DOUBLES KINGS,PRoB-0.5

PLÂYER II

PÀIR JACKS,PROB=0.3DOUBLES ONLY ON ANEXTREMELY STRONG HAND

FñI|ry]

rENs' PROB'o.

PArn 1ENS, PROB.o.7

DOI'BI,ESPAIR PAIR JACKS,PROB-O

DOTELES ACES,PROB'L

PAIR FIVDS ,PROB.0.5DoI,BLES JACKS,PR0B.0. I

141.between the analytic solutlon and the strategies adopted by

experleneed players, as recorded 1n books on the subJect,

(6,8,30).

By studylng flgure 12 the followlng polnts may be

notéd.

(a) inlhen to play flrstThe mlnlmum hand required before a player w111

enter the game, glven that no other player has yet anted, 1s

found from flgure 12 and glven 1n table 17 below.

b) c( o s under whlch the las ?

drops " pJays or doubles.

Generally the last player contlnues to play lfhe has a hand at least as good as the expected hand of the

flrst player to make an ante. The last player then gener-

a1ly doubles holdlng:(1) Jacks or better lf 1 other player ls ln.

(11) Klngs or Aces 1f 2 other players are In.(111) Only an extremely strong hand 1f more than 2 players

are 1n.

(c) Expectatlons of varlous players

Iable 15 shows that players have the followlngexpectatlons (of wlnnlngs) tn the 4-PG.

Player12

34

Expectatlon0. 1130.1290.169

-0.411

t42.

TABLE 17

MINIMUM HAND hII{F:N FITRST TO PLAY: THEORETI CAT. RF:.STIT.T

Mlnlmum hand requlred

palr of 10fs

palr of I fs

palr of 5ts

n. number of olaversto follow

3

2

I

143.

Coffln, (6), makes the followlng general comments about

poker. Flrst, slnce the last player 1s compelled to ante

(while the others are not) ne suffers a dlsadvantage. Thls

ls also evldent in the solutlon because the last player 1s

the only one to have a negatlve expectatlon for the game.

Second, slnce out of players lr2 and 3, the laterpLayers have an advantage over the earlier players, trr that

they have less playerg ahead of them who have not yet anted,

and can thus amlve at a more lnformed declslon. Agaln,

thls effect 1s demonstrated 1n the expectatlons of the 3

players.

The most strlklng aspect of the solutlon 1s the

degree of disadvantage lncurred by player 4 in havlng to

make an ln1t1aI compulsorY ante.

5 .5.4 comparlson of analvtlc solutlon with the strategles

used by experlenced players

over many years certaln strategles have evolved for

the type of game consldered here (see (618,30)). Even

though the rules of some of these games may dlffer sl1ght]y,

the sallent polnts of these strategles remaln largely ln-

varlant, and are 11sted below. These practlcal strategies

w111 then be compared to the theoretlcal strategles found.

here.

144.

(a) When to play flrstIt 1s generally agreed, (618) that the mlnÍmum hand

requlred when flrst to play, 1s as glven ln tabre lB below.

ït may be seen that table 18 and table LT (theoret-

lcal results) agree for n = 2 and 3, wlth the exceptlon ofthe mlnlmum hand requlrements for the second tast player(n=1) . I'Ihereas, 1t ls generalÌy reconmended that any palris sufflcient under these condltlons, the theoretlcal re-sult shows that a palr of 5ts ls requlred. Thls dlscrep-ancy may have the foIlow1ng explanatlon.

F1rst, lt ls posslble that the generally recommended

strategy 1s not correct ln thls lnstance. Certalnly, the

concepts of good play 1n poker have changed over the tast100 years (see (B)).

Second, table 18 ls quoted ln varlous books as not

only applylng to the partlcular verslon of the game solved

here (3Or3Z¡, but also to a verslon of thls game where the

second to last player makes a compulsory ante, half the

slze of that put 1n by the bllnd. Obvlously, 1n thls case,

the second last prayer would play on a somewhat weaker hand

as he ls already partlatly commltted. Thus poker books

generally conslder that s11ght varlatlons such as thls inthe rules do not affect the optlmal strategv, and that the

same strategles may be used 1n slmllar types of game. Thls

ldea appears correct when the flrst and second prayer 1n the

4-PG are consldered, but seems to break down for the secondlast player.

l-45.

TABLE 18

MÏNTMUM HAND }üHEN F'TRST T0 PLAY: PRACTTCAL CRITERIA

o -92

0.90

0.86

0. 80

o.7r

0. 50

palr of Aces

palr of Klngs

palr of Queens

palr of 10rs

paln of Itsany palr

6

5

4

3

2

1

Minlmum handrequlrement

n, the number ofplayers to follow

Correspondlng handx 1n the 4-Pc

146.

Appllcatlon of table 18 to the slmulatlon of poker

It has been lndlcated ln chapter 2 th.at the resultsof tabl-e 18 have been applled to the poker slmulator. Itwas found that a quadratlc functlon of n,

r(n) = -o.oln2 + 0.13n + o.5o

would closely approxlmate the value of x, (frand strength)

for any glven value of n (number of players to follow).Thls functlon, x = f(n), 1s employed 1n the poker slmulat-

oP, as deserlbed 1n sectlon 2.9,

(b) On p1aylnE after another player

Thls dlscusslon w111 conslder the strategy to be

followed lf anothen player has alreatiy entered the game.

The last playerrs strategy 1n thls sltuatlon, however, w111

not be treated here, but w111 be dlseussed separately ln the

next sectlon.

Books on the subJect, (30r32) agree that players

should have better than the mlnlmum expected hand of the

last player to enter, befone maklng an ante. If the

solutlon glven ln flgure 12 ls examlned, then 1t may be

seen to agnee wlth thls princlple. For example, 1f player

1 enters then he must have a hand of at least a palr of 10rs.

In thls case player 2 w111 only enter lf hls hand y ls at

least a palr of Jacks. Simllar patterns may be found

throughout thls solutlon.

147,

Thls behavlour may be approximately descrlbed uslng

a functlon g(1,) = g, + l(r-,c) where g, 1s the last minlmum

expected hand that was entered, and g(f,) glves the next

ml-nlmum requlred hand to enter. Thls functlon ls used. 1n

the poker simulator (chapter 2).lay by the last plaver

Books on the subject, (30,32) do not state any

speclflc requlrements on the mlnlmum hands requlred by the

last player. Only 2 general observations are made.

F1nst, the last player should only play lf h1s

hand compares favcurably wlth the mlnlmum expected hands

of other players who have made an ante. Thls same crlter-lon 1s seen to hold for the sorutlon to the 4-pG (flgure r2).For example, lf player"s 1 and 2 drop, whlIe 3 plays, thenplayer 4 expects playen 3 to have at least a palr of 5ts.Accordlngly, he only plays on a palr of 6fs.

Second., 1t ls recommended that the 1ast player

should only double when there is a good chance that he has

the best hand. An examlnatlon of flgure Lz shows that thlscrlterlon arso holds for the analytlc solutlon. consequent-

Iy, 1t may be noted that lf player 1 drops wh1le players zand 3 p1ay, then player 4 may assume that player 2 holds atleast a palr of I's, whlle player 3 must have at least a

paln of 10's. Thus playen 4 plays on a palr of Jacks or

better, and doubles only on a paÍr of Klngs or above.

148.

CHAPÍER 6

APPLI CATIONS OF THE l¡iORK CARRIED OUI IN THIS STUDY

6. l. Introductlon

As a sequel to the worl< of thls study, two practlcal

appllcatlons w111 now be glven, whlch lllustrate the lnter-

dlsclpllnary nature of thls work.

6.2. Applleatlon of eame theoretl c method.s to a problem ln

networks

In thls sectlon a game theoretlc approach to a partlc-

ular network problem w111 be formulated. The treatment

presented here w111 only glve sufflclent detall to lllus-

trate the prlnclple underlylng thls new approach. A more

detalled paper (Harrls (16)) on the subJect whlch applles

this method to a problem 1n telephone networks, has been

wrltten, followlng a dlscusslon of these ldeas wlth the

author.

6.2.t. Theoretlcal backEround

The ldeas lnvolved 1n the appllcatlon of game theoretlc

methods to networks may be conveniently explalned by means

of an example taken from Dafermos, (7).

Flgure 13 below shows two tOWnS, labelled node x and

nod.e y. There are 5 one-way roads between node x and

node v, called }1nks . Each llnk can only caryy traff lc

ln a s1ng1e dlrectlon, and the amount of trafflc belng

carrled on l1nk 1 ls called the flow, fl'

149.

FIGUNE 13

IINK ]"

FLOltl fr

AN E)(AMP LE OF A STMPLE NETbIORK

LINK 2

¡'LOW fzII

LÏNK 3

Ft0tl f¡LINK 5

FLOI,ü fs\

LINK 4

FLOW fr

ExI I

Deflne the vector t150.

to be the totaf flow vector

where f = [fr rfz rfs rf+rfr]T.Suppose that for any glven totat flow vector I there ls a

cost ci(f) assoclated wlth each l1nk 1 and assume that

the network is such that:

cr(l) = 4frt + 2f fz + 900fr

cz(l) = 6f z2 + 2f $z + 900f2

ca(f) = 6f ,'+ 3fgf,. + B2of3

c,.(!) = Bfnt + 3fsf¡ * B2of,,

cs(f) = 5fsz + L22o\ (I)

Thls means, for example that, ct(f), the cost of carrylng

trafflc fr on I1nk 1, depends not only oñ, fr, the

trafflc carrled by llnk 1, but also on fz, the trafflc on

I1nk 2. In practice, thls type of interactlon may oecur 1f

llnks 1 and 2 form one two way road, and thus the fLow on

one slde of the road w1II, to some degree, affect the flow

of trafflc on the other s1de.

Next d.eflne d( *,y) as the travel demand from node

x to node y. In thls partlcular network lt 1s requlred

that 120 unlts of flow be transported from node x to node

Vr and 120 unlts from node y to node x. Thus

d(*,v) = L2o

d(y,*) = 120 (2)

Slnce, from flgure 13, total flow from node x to node y

ls fr + î? + fs and from node y to node x 1t 1s f2ff¡,

then eq.2 becomes

151.

f1+fs+f5=12O

fz + fL = 12O ß)

where f¡ >

The problem is to apportion the floÏ'i vector € in

such a \ryay that while eq,.3 1s satisfled., the costs associ-

ated. with the network, given by eq" 1 , are in some sense

mininized.. Two crlteria of minimization which are common-

1y used. (see Potts, (zB) and. Dafermos, (Z)) ,are given

þe1ow, while a nerJv' game theoretic approach, will be

d.efined in the next section.

(t) @atlonsystem optini zation consists of mlnimizirtg the

functi on

(4)( )fc ,3"", {[)

(z)

and- therefore gives the lowest possible overall

cost, while ensuring that the constraints are

sati sfie d..

User optimizat,{on

Dafermos, (l), states that for the problern being

considered. here, if each unlt of flow ls con-

sidered. to be an ind.lvidual car, and'the cost

per unit fl-ow represents average time taken to

complete the journey, then the user optlmized'

pattern wil-l occur if each incLividuaL is free

to choose his or,un route lndepend-ently. It is

r52.

assumed that he does thls 1n such a way that

h1s own travelllng tlme 1s mln1m1zed. A

mâthemâtlcal formulatlon of thls crlterlon

w111 not be glven here as lt 1s not relevant

to thls study, but lt maY be found 1n

Dafermos, (7).

6.2.2 Game theoret 1c network oPtlmlzatlon

A new crlterlon of mlnlm:-zatIon, whlch proved bo be

of theoretlcal and practlcal importance (see (16)), 1s

proposed 1n this section.

Conslder the network glven ln figure 13 as a

2-person non-cooperatlve game, deflned ln the followlng way'

Player I must transport 120 unlts of flow (d( * ,v) = 120)

from nod.e x lo node y. He 1s free to dlrect thls

flow in any way he chooses wlthin the llmltatlons of the

constralnts, along l1nks 1.. 3 and 5, Player 2 must l-lke-

wlse transport :.2o units of flow from node y to node x

(Qy,x) = 120). Agaln he 1s free to dlrect thls flow ln

any way that he d.eslres along links 2 and 4. Thus accord-

lng to earlier definltlons e the cost to player 1 for any

glven flow conflguratlon f 1s given by cr(l)+ca(f)+cs(f),

and player I seeks to mlnlmlze thls cost, and hence maxlmj-ze

h1s payoff functlon' Pr(f), where

pr (f) = -cr (f) cs(f) - cr(f) (5)

t53.

Pz(f),Slm|larly, player 2 seeks to maximlze hls payoff functlon,

where

Pz(f) = -c2(f) cr(f) (6)

Hence, the 2 players are ln a competitlve game

sltuatlon slnce the costs sustalned by each player are not

only a functlon of their own actlons, but also depend on the

actlons taken by the opposing player. This game was solved

by means of the LAH-algorlthm (see tabl-e 19) and the solu-

tlon also found algebràiea}Iy by Harnls (aeta1ls w111 not

be glven here as the algebralc method used to flnd the

solutlon was the same as that glven ln 5,5) .

6 .2 .1. Dlscusslon of results

The three solutlons to thls problem, uslng the 3

dlfferent crlterla of optfmlzatlon, are dlsplayed 1n table

19. The fol-lowlng polnts should be noted.

(1) System optlmizatlon provldes the overall

mlnlmum total cost, c(f).

(1j_) Game optlmizatlon provldes a lower total cost,

e(f), than user optlmlzatlon' Although 1t

has not been proved lt 1s conjectured that thls

may have the followlng explanatlon' In the

game theoretlc optlmizatlon there are two

confllctlng players. However, the deflnltion

of user optÍmlzatlon, as gÍ-ven earller, means

that each of the lndlvtdual- users ls ln

1tr

TABLE 19

RES FOR NETüIORK OPTII'{I ZATTON

Deflne the total cost c(f) =5x cr(f)

c([) 317 ,5oo 318 ,300 322,00O

fr1¿

ál3

õIT

4I5

50.2

66.t

35.2

53,9

34 .6

54.5

66,2

4o .6

53.8

2t1.9

61. 3

64.3

47.9

55.7

10. B

SYSTEM

OPTIIVIIZED

GAME

OPTTI,IIZED

USER

OPTIMIZED

r55.

competltlon with every other user' Thls

greater degree of competltlon ln the user

optlmlzed system would tend to create a lesser

degnee of eooperatlon, and hence hlgher eosts.

Thus, lt has been shown that game theoretlc methods

can have appllcatlon ln other, not dlrectly related, areas

of applled mathematlcs, and the numerlcal technlques

developed 1n thls study (e.g. LAH algorlthrn) can be applled

to such problems. In eoncluslon Harrls, (t6¡, has shown

that the game theoretlc optlmlzatlon prlnclple has lmport-

ant theoretlcal and practlcal appllcatlons 1n the theory

of telephone networks. It 1s consldered by Harrls that

the game theoretic prlnclple can be used to clarlfy centaln

heurlstlc methods whlch are used by telephone network plann-

1ng authorftles. Also, the game theoretlc solution may be

used as a close approxlmatlon to the system optlmlzed

solutlon.

6,3. Buslness model

As has been observed ln chapter I, even though prev-

lous wrlters have remarl<ed on the slmllarity between poker-

Ilke games and problems of buslness operatlons, (29), no

exp]lclt examples could be dlscovered ln the llteratune'

Thus, lfi thls sectlon, a buslness nodel ls constructed to

parallel the 4-PG solved ln chapter 5. The broad prlnc-

tples on whlch the model ls based are as follows.

156.

Conslder a situatlon lnvolvlng 4 competlng manufact-

urers denoted by !r213 and 4. Each manufacturer must

declde whether or not to market a partlcular product and

thereby involve hlmself in a sltuatlon 1n whlch he w111

elther make a proflt or a loss dependlng on varlous assump-

tlons, whlch are glven beIow.

(a) The flrst assumptlon 1s that a hlghly speclallzed

product that has a l1m1ted appeal is belng marketed.

The manket has the property that lt w111 ultlnatelybuy from only one of the manufacturers. Three

practlcal examples of such a sltuatlon are:

(1) Vühen the market conslsts of only one customer

for example, a government department.

(11) When the market 1s so smalI that eventually only

the most successful manufacturer will be able to

operate 1n 1t. The remalnlng manufacturers wll1

be forced to cut thelr losses and qult.(fff) Wfren a monopoly situatlon can be establlshed by

one of the manufacturers.

(¡) The second assumptlon 1s that when the wlnning manu-

facturer emerges, hls total proflt wltl equal the

amount spent on advertlslng and promotlon by the other

manufacturers who have entered- thls competltlon. Thls

assumptlon may be Justlfled 1n the following way.

r57 .

It 1s known that 1n certain situatlons, sales, and

therefore proflts, may be dlrectly related to ad-

vertislng expendlture. If the maJor non-returnabl-e

cost lnvolved 1n thls buslness 1s that of advertls-

ing, and tf all the advertlslng ls such that ltbeneflts all manufacturens equivalently (f .e.

develops new sectl-ons of the market whlch w1II

eventually all be servlced by the f1nal wlnner)

then 1t 1s posslble that proflts w1II equal totalamount spent on advertlslng. A historical example

of thls ls glven by t'Ilnkler, (36), and coneerns the

devetopment of the tobacco lndustry ln the U.S.A',

around 1900, whlch was eventually domlnated by one

company. Durlng cerbaln periods of lntense comp-

etltlon whlch preceded this, l1ttle or no proflts

vûere made, but huge markets were developed by allmanufaeturers as a result of the priee cuttlng and

heavy advertlslng whleh took pIace. FlnallÏr a

slngle company vras suceessful 1n obtalning a mon-

opoly of over 90% of all the tobaceo trade, and

eventually made enormous proflts. It would not be

unreasonable to assume that these proflts were

dlrectly related. to the huge sums spent 1n advertls-

lng and promotlon by all manufacturers during the

perlod of lntense competÍtlon.

(c)

rs8 '

Each manufacturer, before he d'ecld'es whether or not

to enter the market, needs to rate h1s product 1n

some way. tr'or the purposes of thls model 1t w1II

be assumed that thls ratlng ls glven by some number

x, 0 < x 3 l, where thls value has the follow1ng

propertles.

(1) A clear and common understandlng of thls

valueexlstsamongallplayers'Thatls'each manufacturer, lf he had to value any

one of the products, would glve it the same

value as all the other manufacturens ' Thls

would, 1ri practice, arlse because those who

conslstently over-valued or under-valued

tLreir products would go out of buslness ' and

only those who could' glve an accurate value 'would remaln.

For example pawn-brokers and used ca?

salesmen must exerclse thls ablllty' wh1le

new car manufacturers must also learn to make

accurate, conslstent value Judgements of thls

tYPe'

(11) By conmon agreement thls value x has the

folIowlng meaning' Thls 1s best lllustrated

by an example. Consld'er the case of a used

car salesman who speclallses ln appralslng a

(d)

r59.

partlcular model of car. Because of hls

experlence he is able to rate any given ear'

l-n relatlon to all other ears of that type 'and. state that thls car 1s better than say

p7, of cars of this type . Then, 1rI thls case,

deflne x to be equal to P,/100 '

It J-s assumed that one of the four manufact-

urerslspreparedtolnltlatethecompetitlvesltuatlon 1n the follow1ng üray. He postu-

Iates that there may exlst a need for some

partlcuLar producb. Although he knows that

he w111 be able to manufactune 1t he does not

yet know Just how good h1s product w111 be

(1.e. he does not know hls own value of x)'

However, he 1s prepared, ãl the cost of 1 un-

Lþ, to mount a small advertlslng campalgn'

whlch w111 establlsh whether thls partlcular

product w1lL 1n facb find a market ' At thls

polnt the other three manufaeturers are ln-

formed of the results of thls survey, and

thus the competltlve situatlon 1s created'

Theotherthreemanufacturersrknowingthelrown value of x', must now make a decislon

whether or not to enter the market ' Thls

w111 lmmedlately lnvolve them 1n a cost of 2

(e)

(f)

160.

unlts for advertlslng. It ls assumed that

the competltors have certaln flxed pollc1es

establlshed. over a perlod of tlme, that

declslons are made ln some deflnlte pre-

determlned order. Condltlons are such that

all are aware of any declslons whlch are made.

I,rlhen all manufacturers have made known thelr

lntentlons, the last manufaeturerr now also

knowing hls value of x, haÊ 3 cholces.

(1) He may choose to drop out and forfelt

hls lnltlal outlaY of I unl-t.

(11) He may d.eclde to contlnue, but on equal

terms wlth hls rivals. Thus he lncreas-

es hls expendlture by 1 unlt to niatch

the total amounts outlayed by the others.

(111) ff hls product happens to be partlcular-

1y good, and because of hls good strat-

eglc posltion (1.e. he was flrst lnto

the market and has had the greatest

amount of time to consider the sltuatlon,

knowlng the actions taken by the other

players) ne may lncrease h1s outlay by a

further 3 unlts. Thls w111 lntroduce a

tendency for hls competltors to match

hls expendlture (at a cost of a further

161.

2 unlts ) as they are alreadY fu1lY

commltted and must contlnue. Not to

match this inereased expenditure would

fnean that they would certainly lose.

(g) The winning manufacturer 1s determlned ln the

followlng way. Inltla1ly, each manufacturer

computes the value of x applylng to hls own

product. Denote these values by x¡ ,x2 rx3,

and x4 respectively. At this stage none

of the manufacturers know the xi values

calculated by any of their rivals. There ls

some flxed chance¡ Q, equal for all compet-

itors, of maklng a last mlnute technologlcal

breakthrough, whlch, ls of such a magnltude

that, 1f achleved, it w111 ensure that partlc-

ulan competltor of wlnnlng the market. How-

ever, lf two or more manufacturers achleve a

slmultaneous breakthrough, then the wlnnen is

the one wlth the higher lnltlaI xi. If no

technological advances are made then the hlgh-

est xi w111 w1n.

8q.1, chapter 4, deflned the functlon Tn(x) where,

glven some random number r, 0 < ? I 1, then

I xiq<r<rn(x) = l^- [2+x;0<r<q

]-62.

Thus, uslng the above deflnltlon lt follows that the wlnner

may be deflned as follows. Each value of xi ls replaced

by tn(xi), and, then the hlghest value wlns'

Now, uslng (a) to (e) above, 1t 1s posslble to

formutate a buslness operatlon whlch completely parallels

the 4-PG. Famlllarity with the ll-pc and polnts (a) to (e)

above, will be assumed 1n the ensulng dlscusslon'

6. 3. 1. Deflnitlon of buslness model

Supposethatfourmanufacturersdenotedl,2,3and

4 are ldentlfied wlth players L,2,3 and 4 ln the 4-PG. In

the ensuing dlseusslon the 4-pC w111 be descrlbed and

lnterpreted ln terms of the buslness model '

Flrst, player 4 makes an ante of 1 unlt ' Tttls

corresponds to manufacturer 4 ln1tlal1zlng the competltlve

sltuation by outlaylng I unlt as described ln (d) above'

Next, the four players 1n the ll-PG are dealt hands xI rX2 r

X3andXr+.fhlscorrespondstothefourmanufacturerscalculatlng the goodness of thelr products 1n terms of the

variables x L tx2 txs and, x4 as described ln (c) ' Now'

beglnnlng wlth player 1 each player, lri turn, decldes

whether or not to enter the game by maklng an ante of 2

un1ts. Thls same sltuatlon occurs 1n the buslness opera-

tlon (see (e)), wlth each manufacturer, 1Ê turn, declding

whether or not to enter the competitlon at a cost of 2 unlts '

163 .

Player 4 may novl drop out, play ohr or double'

thereby forclng hls eompetltors to doubte. Thls also

occurs ln the busl-ness mod.el- and ls descrlbed 1n ( f ) ' where

manufacturer 4 also has the option of dropplng out (and for-

feltlng I unlt), remalnlng 1n the market (at a total cost of

2 unlts), or d.oub11ng hls expendlture (to a total cost of 4

unlts). At thls stage of the 4-PG all hands xi are

lmproved by the transformatlon Tq(xi) as glven ln eq.I

chapter 4. Exactly the samê, procedure ls used 1n the

buslness model and thls 1s descrlbed 1n (g)' The wlnner

1s then ld.entÌfled ln the same way ln both cases. The

4-pG allows the wlnnlng player to take all bets made, whlle,

the business moctel allows the wlnning manufacturer a proflt

equal to the sum totals of all amounts spent by hls compet-

ltors, as descrlbed ln (b).

Hencerthesolutlonfoundforthe4-pCmaybe

dlrectly applled to the buslness mod.el, and each of the

four manufacturers can conslder h1s value of x1 (goodness

of product) to be equlvalent to a hand xl 1n the 4-PG,

and take actlons in the buslness sltuatlon corresponding to

the ones +.hat he would' take 1n the 4-PG'

6.3.2. Dlscusslon of buslness model

The dlscusslon above showed how a buslness operatlon

could be deflned to parallel the 4-PG 1n such a way that the

optlmat strategy for the {-pO could be applled to the busl-

164.

ness model. Thls suggests that othen buslness models of

thls type may also be studled and solved by means of the

theoretlcal methods developed ln thls study. The above

example also lndlcates, ln a practlca] way, the close

connectlon between buslness and poker.

]-65.

CHAPTER 7

SUMMARY, CONCLUSIONS AND

DISCUSSION OF NEhr hrORK

7.L Summary

This sectÍon w111 summarlse and dlscuss the work

carrled out ln thls stud.y. The lnltlaL j-nvestlgatlon of

the problem of solvlng poker-I1ke games, revealed the

followlng polnts.

(a) A revlew of the llterature showed that poker

uras amongst the most dlfficult of card games, and

that a signlflcant amount of work had. been

carrled out 1n thls area by Bellman, Friedman,

Kar1ln, von Neumann, Restreppo and others.

However, the ganes solved could not be applled to

any commonly pfayed varlety of poker. Thls was

because the problems posed by poker-Ilke games

were sufflclently formldable to restrlct the

workers noted above to relatlvely slmple analyses.

(b) Apart from the v¡ork of F1nd1er, no treatment of

poker simulatlon was noted ln the llterature.Thls was surprislng because:-

(1) Slnulation of poker offers a worker

the capablllty of evaluatlng optlmal

strategles and of testlng the va1ld1ty

of theoretlcal resul-ts.

L66.

(if ¡ A poker slmulator coul-d be adapted topfay lnteractively wlth human opponents.

Thus slmulatlon appeared to offer attractlveposslblllt1es of glvlng further lnslght lnto the theory

of the game of polrer.

(c) A revlew of the llterature showed that the maln

dlfflculty 1n applylng game theoretlc methods to

poker arose ln the solutlon of the resultantequatlons. It followed that the appllcatlon of

numerlcal methods mlght yleld useful results.However, thls approach had not prevlously been used

for poker, and no sultable algorlthms u¡ere found

to exlst. It was probable therefore, that the

numerlcal approach nlght requlre much new work.

(d) Flnal1y, desplte comments by varlous wrlters (L9,29)

on the connectlon between poker and buslness

operatlons" no appllcatlon of poker resuLts to a

practlcal problem, could be found. Thls suggested

that an lmportant sequel to a bheoretlcal- study of

poker should be an appllcation of theoretlcalresults to relevant problems.

The foregolng consideratlons Led to the work of thlsthesls belng carrled out 1n four maln parts aB follows.

167 .

.2 ach to the blem OS s thes s

(a) Slmulatlon of poker

The flrst part of thls work, see chapter 2, dealt

wlth the simulation of poker. Durlng thls research the

followlng ,,c gults tüere obtalned.

(1) probabltltles of wlnnlng wlth poker hands

were deflned and calculated

(11) values were obtaíned for computatl"on tlmes

for the slmulatlon of Poker

(fff) a rnethod of determlnlng optimal strategles

uslng slmulatlon was presented but was shown

to be lmpractlcal, as too much computer time

was requlred

(fv) the slmul-atlon program was modlfled to play

poker lnteractlvely. However, thls part of

the work could not be carrled through to a

satlsfactorY concluslon because :

1. sufflclently long poker sesslons between

the machlne and human players could not

be arranged

2. poker needs to be p1e'- yed for money 1f the

results are to have practlcal slgnlflcancet

but thls could not be arranged 1n the

clrcumstances of the work of thls thesls.

168.

, As a result of the work done 1ü was concluded

that slmulatlon offered a posslble method of solutlon

bUt there v,rere Conslderable computatlonal dlfflcultles

lnvolved. Aceordlngty, a prellrtlnary study was made

of a game-theoretlc approach and thls appeared to

offer good prospects of achlevlng a solutlon more

easlly than slmulatLon. As a result the slmulatlon

studles were dLscontlnued.

(b) The Eame theor etLc ârlNPôâ. ch for so'l vl no eâmes and

numerlcal methods

The game-theoretlc approach to the study of poker-

11ke games requlred the solutlon of certaln compllcated

equatlons. Research showed that these equatlons could

be most convenlently sotved uslng numerlcal methods'

but a survey of the llteratr¡re dld not reveal any

appllcatlons of such methods to poker-}lke games.

Hence, much work was carrled out ln thls area, âb

1nltlo. This work ls descrlbed ln chapter 3 and forms

an lmportant part of thls studY.

The maln results of thls work are as follows'

(1) Two numerlcal algorlthms capable of solvlng

poker-llke ganes were prograrnmed. The first

of these was a modlflcatlon of Rosenfs method

(31). However lt was demonstrated that thls

169.

algorlthm was too slow for the purposes of this

( il)

work.

consequently a new lterative method called the

lookahead (LAH) algorlthm was formulated'

Thls operated on a lookahead pr1nclpIe (as

descrlbed 1n chapter 3) and proved to be three

tlmes faster than Rosent s modlfled method'

(c) Solut of realist c poker-Ilke games

chapters 4 and 5 descrlbe the work carrled out

lnapplylnggametheoreticmethodstoaco¡nmonlyplayed

varlety of poker. Thls research was structured as

fol]ows.

(i) Slnce the game considered was too com-

pllcated to treat ln 1ts entlrety, the rules

were carefully simpJ-lfled' ln such a way

that much of the essentlal character of the

game remalned unaltered.

(11) In the course of derlving payoff functlons

for thls game a muLtl-dlmensional lntegraltwhich commonly arlses 1n games of thls type'

was encountered. Its eval-uation for the

n-person case presented conslderable

d1fflcult1es and merlted treatment 1n a

separate appendix (Appendlx A).

170.

(111)

Thls result played an lmportant part 1n the

method used for solvlng the 3 and 4 person

poker-llke games, and could be extended to

poker-Ilke games lor 5, 6 and 7 players.

Next, the lookahead algorlthm was applled

to the payoff functlons and the game solved.

This solutlon had several new features.

1. Prevlous research 1n thls area had only

considered poker-I1ke games wlth no more

than three players. '''-:'oy were not

sufflclently reallstlc to have any broad

practlcal appllcatlon to real poker wlth

more than two players. In thls study, by

uslng the resutts obtalned durlng the work

on poker slmulatlon, lt was posslble to

relate the solutlons obtalned to a real

game of poker. It was then found that

the theoretlcal optlmal strategles agreed

wlth the strateEles recommended by . rr¿ttÀ.{}{" trLvy, ¿.r.rv{, çt\ Ños. 4À;¡1¡.tu."a ';u¿l-os r.;*u* l"a^ð w1u'wÜoaffifffäå"'êå"';Iåffii;

/, 30 ) . For exampre ,

mlnlmum hand requlrements, when flrst to

enter, agreed closely wlth the work of

Reece and ldatklns , ( 30 ) . AIso the

dlsadvantage of playlng last was demon-

strated analytlcat1y, and lt was conflrmed

LTl..

that players closest to the last player

had the hlghest expectatlons.

2. Comþuter runnlng times requlred to deternllne

the solutlon were recorded, and lt was noted

that, usfng present methods, too much

computer tltne would be regulred', 1f more

than 4 person games were to be solved'

(d) Practlca1 aopltcatl-ons of thls work

Two appllcatlons of thls study to practlcal

problems are presented 1n chapter 6' The flrst

appllcatlon descrlbes a game theoretlc approach to a

problem 1n networks. Thls work deflnes the network

to be a game between 2 ot more players. ThlS gane can

then be solved by the methods glven 1n thls study.

The results obtalned. by thls approach have proved use-

ful ln the analysis of telephone networks ¡ âs they

d,eflne a new crlterlon of optlmallty, (16) '

The second appllcatlon deflnes a business

operatlon lnvolvlng four manufacturers, who must

declde whether or not to market a neür product. Thls

buslness operatlon 1s dr{;¿ol \- pa",.1l"t .,,-.,:;.,.+.,i'.,-r.('"

'" the 4-person poker-llke game. Thus an exact

crlterlon of optlmallty that can be applled by each

manufacturer when dec1d.1ng whether or not to enter the

market follows from the solutlon to the poker-l1ke game'

L72.

Furthermbre, thls soLutlon also lndlcates to each

manufacturer the amount of h1s expecteö proflt.

7.3 Further areas of research

(a) Simulatlon

As a result of the work carrled out on the t

slmulatlon of poker, lt appears that further research

could be dlrected 1n the folLowlng areas.

(1) In chapter 2 a theoretlcal method ls descrlbed

by whlch slmulatlon could be used to determlne

optlnal poker strategles. But experlmental work

showed that thls approach was computatlonally too

slow. A nevù approach to thls problem would be to

study the }og1caI processes by which an

experlenced poker player analyses the game, and

ls able to adapt h1s strategy to glve hlmseLf the

best results. Perhaps, as a result of such work,

fewer games would need to be slmulated to determlne

optimal strategles. Thus the total amount of

computatlon would be reduced and lt would then be

feaslble to carîy out these calculatlons, on a

computer, ln a reasonable t1me.

(11) U1tlmately, the success or fallure of any poker

playlng program can only be measured by play

agalnst human opponents. Thls can best be canrled

out by uslng lnteractlve poker programs under

LT3.

reallstlc playlng condltlons. Money should be

used 1n such experlments, and the poker sesslons

should extend over many hours. Resul-ts

obtàlned from such experlments could be

analysed. to glve further lnfofmatlon about þoker'

to determlne weaknesses 1n the program and üo

lndlcate ways 1n whLch the program log1c mlght

be lmproved.

(b) Numerlcal methods of solvlne Aames

Slnce lt would be of great practlcal lmportance

to solve more complicated games, new research ln the

area of numerlcal methods of solutlon should be carrled

out. One polnt of departure would be to carry out

further work on the Lookahead algorlthm. For examplet

the varlous comblned effects of alterlng the degree of

lookahead, the ratlo between successlve step slzes

and varlous other parameters of the algorlthm could be

studled. In add1tlon, completely new approaches to

thls problem, could be attempted. A posslble source of

new ldeas lies ln the analysls of the human' lntultlve,

approach to the solutlon of games, t¡y whlch glfted

manAgers achleve favourabLe outcomes |n sltuatlons of

great complexlty, ( 35 ) .

]-7\.

(c) Solutlon of more comollcated eames

It would be feaslble to solve a J-persÔn poker-

I1ke game, bV game-theoretlc methods. There are'

however, two obstacles.

The flrst dlfflculty 1s the large number of

strategy varlables (many hundreds), whlch would arlse

1n the analysls of thls game. New work may dlscover

means of representing the strategles uslng a smaller

number of varlables.

The second dlfflculty arlses ln solvlng these

games. Obvlous}y, lmproved numerlcal methods, w1ll be

of assistance. But, another app:-'oach mlght be to devlse

a technlque which uses the algebraic method of

solutlon, solved on the computer, by the methods of

symbollc manlpulatlon.

(d) Appl1 ed qames theorv

Game thei.,.; ccii¡ Je apririled to a wlde clrcle of

problems. UntiL nol^J, as e consequence of the ..-tifflculty

of flndlng solutlons, wopk in thls area þas met v¡lth

tlttle pracblcal success. However, work cr''

appllcatlons shouLd contlnue, because any progi:ess

would be oí" pi,acticaL use, as þas lreen particularly

demonstrateo ln chaPter 6.

L75.

In partlcular, the new work on the game-theoretlc

approach to netwörk problems has ylelded useful

results and work ln thls area 1s belng contlnued by

another researcher.

176.

APPENDIX A : EVALUATION OF PAYOFF IUNCTION INTEGRAI,

.A..1 . Introd.Jrctl on

It was found. thrat, in the course of this stud.y

(see chapters 3r4r5), a certain integral arose repeated.ly.

An intensive search of the literature revealed. no publlca-

tion deallng withr this or any other related. integral.

Accord.lnglv, a method. for tþe e.¡aluation of this integral

was d.eveloped., and. is presented. in this appencLix. This

integral is d.eflned- j-n the foll-owing way'

Let

Xï'¿(*nr"'rx1)(w;Lrt

bn

tJ

Xn =ân

i=1i/nlf xn>

ifxn<for allfor any

¡ ' o rll (t )X1X1

then the required. lntegral is

lLbbn b1 ... I

xL=ãL

xH '¿ XH r¿ (*n, ., .x1) dx1. . .dx¡

(z)â¡ at

where â1 ( br for i=1 ,. . .o.

Thls lntegral wil-I be evaluated. in the followlng way.

Flrst, the case rt=Z will be consid'ered'' It w111 be

shotrn that ttrere is a relatively straight forwarcl, graphic-

al nethod. whlch can evaLuate this integral, and. whlch, ifi

theory, could- be generalised.. However, it will then be

proven that it is not practically possible to extend- thls

inethod. to the general case ("rty value of n)'

IT7 .

Accord.ingly this lntegral (for n=2) ls evaluated.

by using a more cLifficult a1d. less obvious technique, uhich

carLe however, with some hard.shlp, be generaliseil to give

the result for arþitrary values of rlo The valld.ity of

the etæression f or,¡nd. is then checked. by comparison with tTre

results from a monte-carlo nethod., which is ltselfr howevert

shown to be too slow and. inaccurate to be used. here.

^.)- l'ìq'l nrr'laf.lnn nf inf.core] fon n-2

The lntegral will be calculated., for the case rt=2c

by two method.s. First, by a graphlcal method., and' second-t

by an algebraic nethod..

4.2.1. Calculatign of 1nlegrat for, 4=2 bY-a qraphical

method.

lf tt=Z then €esc 1 and. 2 take the form

xU'¿(*ry)x> yx<y_fw-L¿ t

,ifif 3)

and.

xE'¿ (4)

X=â Y=C

ConsüLer the case c <

figure.A,.1 below illqstrates the d-omain of integration.

t: :l f f xH,¿ (xry)dxdy

u8.

FIGT'RE A.1 DiagonalvA Y=x

I

d

x

The d.ouble integral is taken over the quad.rilateral PQSR'

However, inthat part of PQSR which is below the line

ll/Y, x. > y, and. hence, from eq,3, xU'¿(*ry) is constant

over ttris reglon, ârrd equals 1M. Similarly, Xä'¿ (xry)

is constant over the region insid.e TVIISR and. equals l.Accord.ingftrr ,

ca

1a

X=â V=C

area lnsid.ePSSR

area insiilePQWT

xU,r(*ry)dxdy

"LU,¿ (*ry) dxdy

xU,t (*ry) d.xd.y + xä,¿ (*,y)dxdy

fftl

tftt IIarea ins id.e

ÎVIISR

w d.xd.y +

sid.e

t d.xd.y

sid.ePQliyT TUISR

r79.

= t^l (area inslde PQI/íT) + 9" (area lnslde TI¡ISR).

By the geometry of flgure 4.1 lt may be seen that:-

area 1nslde PQI,rlT = *(b-a)¿ + (¡-a)(a-c)

anea 1ns1de TI¡ISR = å(b-a)2 + (¡-a)(d-b)

Thus

(5)

(6)

xT'sbd

ac

d

1. O<aSc; OlbSc; c

dI

2. 03ale; clbscl; c

3. 03alc; cl<b<l; <t

l

t: :l*T" =r,(b-a) ( a-c)

l =w[ * (u-a) 2+ (¡-a) ( a-c ) ]+r, I I ( ¡-a ) 2+ (¡-a ) ( d-b ) ]

There are, ln all, 6 posslbllltles of the above type'

and these are enumerated below along wlth the correspondlng

values of

',rw , ßr\2 tbd

ac

ab

a

t" :]ïÏ' r,=sr. *(t-a) 2+1,[ (¡-a) ( ¿-c)

-l(b-a) 2 l

c

a

b

*ï'ol-o ul-rt

*( a-c)2+(a-c) (u-a) ll_a cl+r,[ å(¿-c)2+(¿-c)(c-a) ]

180.

h. clastl; clb<tt;

5, c-<actl; tl<b<l

=r[*(¡-a) t+(¡-r) ("-"

+r,[ å(b-a) 2+(u-a) ( ¿-b)1)l

d

c

dl.

c

t" :]a b

ab

#.,1

*T'u

*T,nl-o ul=rtå(d-a)21

- [" "l**¡(b-a)(d-c)-å(d-a)2]

6. t[<a<l; tt<b<I

cl

c

b

The above results may now be employed to evaluate

thls lntegraI, and a fortran functlon was wrltten to do thls.

However, 1f this method of evaluatlon 1s applled to the

lntegnal wlth n=3 then lt may be seen that 90 cases arlse,

and 1t would be extremely dlfflcu1t to tabulate all of these t

let alone attempt to apply thts method. to the case when n=4

or more. Hence, as lntegrals of dlmenslon 3 and hlgher

must be evaluated later, ão alternatlve method 1s developed

1n the next sectlon.

a

A"2.2" C â l eìrl,ì tion 'inteornl for n-2

181.

bv en a]-sebraic

methoit

An algebraic rlethod. for evaluating Xä ¿

H,t (*,y)dx dy

t: :lwil-I now be given. This rnethod. is based. on the icLea of

expressing the integral as the sum of l+ separate sub-

integrals, each of utrlch may be simply evaluated'.

First, wrlte-l¡ d. I .b Fx

xä'¿l_u "J = I I xu,'¿ (*,Y)d'xdY+ fiïNow d.efine

Thus, by eq.8

and. hence

x=â l-xSubstituting eqs.8 and. 9 into eQ"7

X=fl 'f=C fi=fl $=X

Èl¡xl^bFrc

"Ï '' L' "l = I I xl'¿ (*'Y)dx dY

X=â Í=C

Q).

(s).

(g)

ÉIt xl þ ,*

"î''L' .l = I I x\"(*'Y)dx crY

X=fl g=d-

5lb xl "b d-rî''L" .l = I J *\,1(*,y)¿x ay

xä' Lbd.ac = J\'tt l

b

a

xcI -ri"[: ä]

In ord-er to eval-uate J L ilef ine

(to)

182.

g = nax(min(b,c)ra) (tt).

As will beeome apparent Later this particular choice of g

has been mad.e becairse, since a(b aniL c(d. (see eq,.z) it

follows thatt-( i) 1f c € [ ",b]

then B=c

( ir) if c < a then B=a

(iii) if c>b then B=b

Next expand eq-.8 by intnod.ucing g into the 1im1ts of

lntegrationx

X-â $=G

b

a cJI'¿

JT'¿bxac

K\, ¿

KH '¿

XE,¿ (*,y)dx dy

fry,l, ¿ (*ry) d.xd.y+ xä,¿ (xry) d.xd.y

x-$ f=cX=@ )r=C

and. let

and.

[; :] = f f.u,¿(x,v)dx öv

bx

(tz7

(13)

bg :]

X=â /=C

rr"\,¿ (*ry)¿x ¿y

X=$ $=C

then eq.12 becomes

,ï,,[: :] = Kr,,[: :] + K\,1 gc

18 3.

Finst oonsid.er KT,¿[:

:]

It wl1l be shown that

for the 2 Bossible ceseê cÞa and.

( r) c>a

In tfrls case lt follows from eq.11

thr¡s aà8. Hence as y e. [" rx]

and. so XH'¿ (*ry) = A from eq. 6.

Now

t= ¿ G-a)g+a_c

](=fl }=C

(14)

C( â"

that g - min(brc), and.

and. x € [rrg] then xly.t

KT'¿

K\, t

[: :] x. l

[::] = f f *r'¿(*,v).dxdv

/ dxdy

X=f. $=Q

and. hence eq.14 hol-d.s,

(t) c<a

Again fro¡n eq.llr c(â lmplies that 8=a.

gx

l= r rHence KI,¿ acX=â l=Q

xH ,¿ (*ry) dx dy

= O as the variable x Ìras zeto rarrge'

Thus, again eq.14 hold-s.

Hence eq.14 alwaYs Ìrold-s.

Next,usingtheSamemethod-itwillbeshollinthat

= w(t-g) ÞLg c

18 4.

lc(b and. c>b"

thus 8àc "

then x>Y t and. so

l may be calculateiL,

Kå'¿bxge l (15)

2

Again consid.er the 2 posslble cases

(") c<b

From eg.11 c E = max(ârc),

Hence as y€ ["r*] and. x€ [g'b]

XU,t (*ry) = ur from eq. ¿+.

Thrr^s¡ âs before, KE'x

and. it is founil that

bxgc

Kä, tbxgc

- Kl-'x

l l= w(¡-g) þ+g c2

(¡) c>b

The same argument as ïuas used. earlier rnay be employ-

ecl to show that g=b and. that consequently eq.15 hold-s.

Hence eq.'15 hold.s for all câsêsr

From eq,.13

rT,,¿b

a

x

c l l lgxac + KU,T

bxgc

where from eq.11,

15

g = max(armtn(trc)) and- uslng eqs"l4 and'

bl¡ xl,î,'L: :] = t(e-a) [E- - "] + w(t-e)[V - "]

r85.

b+h c

( 16)

çtt)By a slmilar means 1t may be shou¡n that

if h - max(arrnin(¡ra))

rhen ,l ,, [l äl = ¿(n-a)l-+- - "] * *(r-h)[" Lu -J

l o2

( 1s)

By substituting eqs.16 a¡lct 18 into eq"1O it is found. that

ry,¿bd.ac t= ¿(e-a) g+a

c ]**{r-s)Ff cl2

where

is glven by eq.1!"

4.3 Genefalisation of the algebraic method

The method. usecl above for the case rt=Zc rnay be

extend.ed., with some d-ifficulty, to the general n-d-j-mensional

case. The general approach is to express the integral as

the sum of several integrals, each of which may be

ind.ivid.ually evaluated.. This is carrieiL out in several

stages.

- r(n-a)[ry - "]-"to-")F* - "] (rg)

g and. h are d.efined. by eqs.11 and- 17 nespectively'

Thus, ttre integral dlb d_l

xg,, L " "l

186.

First, the integral. wil1 þe split into zr'-t sub-

integrals, 1n an analogous manner to the CaSe for r=2.

second-Iy, a set s will be d-efined, which may be used' to

e)çpresg more convenlently these 2n-L subintegralsn

Finally, each of tþese subintegrals will be ind-iv1d'ua11y

evaluated to give the final result.

In what fo110ws 1t will be convenlent to d,enote

xH'¿(*nr.. rx1)ô*n-1n.'d.x1 by tb)t.

From eq.2

xH,¿ t

bn þtâ¡o.râ1

bn b1

â¡o.oâ1 Xl'¿ (xr,, . . .x, )dx¡'''dJ('

,r$ +

L

Jtn =Bn

Hence expand.lng tl-is integral-

] =-"{ïl-.

x:-=ãt

n-1XH '¿ oco

- t=ân- 1 X1=ât

vit dxnn-1

aaa

taoc

n

Xn- 1=ân- 1 X1=Xn

n- I n n- 1 ,)

^n-4n Xn- 1=ân- 1 X1=â1 Xn- 1=ân- 1 x1=b1

dxn,1,*rr,

n- 1aaat

I

bn b1

â¡oo.â1

naat

( zo)

Thus ee. 20 may be furtTrer expand.ed- to give

trI\t_

n- 1aoa

,ir, Tn

ItnX

Xn- 1=ân- 1 Ï2=fla X1=â1 X¡* 1=â¡- 1

2

l*=ân

n

X2=Xn Xf=â1

lV)U

XH, I

n-1

a ao

a ao

n-1

aaat*+

n-1a aa

n-1

187.

tt?, dxtt

tt& aaa

X¡* 1=â.¡- L &Z=dZ X1=b1 X¡- 1=ân- t

Jî¿F{- tv

Xn-1=8n-1 X:-=ãL Xn-1=ân-1 ]f'2=b2 Xt=4t

n-1 tyÐ+

t), dxn

taa

Xn- r=an- L XZ=ã2 Jf-¡=b¡ Xn- j-=ân- t Xz=þz X1=b1

By repeating the above process the final result is

bn þ1

â¡o.râ1

naù a

'sÍrt - f"... f" f*'.Xn=ân Xn- 1=ân- L XL=ãt X¡- 1=ân - alf'2=à2Ï-r=bt

tltll

Xn-t=ân- ¡ X.2=b2 Xt=ê1 Xn-t=ân- t :r'z=þz X1=b,r"...r" /"" r'... r" r'

+(-1)n'' (zt)

X¡- 1=b¡- 1 x1=b"

8q.21 nay þe expressed. 1n a more convenlent form, but to clo

this some new notation must be lntroiluced-.

1BB.

Define the set

S = ["n*r. e."ucal where c¡ = a¡ or bJ. (22)

Let the set J be the set of all possible Sr (note that

there are 2n- L d.if f erent S which may be f ortned. as there

are 2 possibilities for each element CJ, and. there arie

n-1 elements in S).

For any Se J d.efine a fi¡nction

p (s)

xl ,

ofbofb=[:

1;1i

if the no.if ttre rroo

elements 1n Selements in S

J

J

evenocLd.

r-gis

(zt)

Def ine YH:Í ("n-r-, o..c1 rxn)

.](- ^X-l"... I XH'l(xnr.e .xr)dx"...d.x¡ttXn- 1=Cn-1 X1=C"

then by inspecting eq.21 it may be seen that this equation

can be ï¡rltten

bnzH'¿

â¡ Cn- 1 . . . C1

SeJ l> p(s)YHrl ("n-" ¡ . . o c1 ¡xn ) ( 2l+)

n

T,

t

nbn b1

â¡oorâ1

*=ân

Now d-efine

lthen eq.24 becomes

X=ân

Yfr¿Í (cn-r e.. rcl rxn)d.xn Q5)

XH, 2

Then

bn b1

â¡r.oâ1 l > p (s) zH,¿SeJ Cn-t.. rC1

l

(*-" r ) dx

d.

An Cn-tr. rC1

[::

1Bg.

+

(26)lIn evaluating

zl'¿

b

â Cn-1o..C1

þn

â¡ Cn-teoeCl

the ord.er of evaluatlng wlth respect to cn-lro..c1 , is

lmmaterial, hence there is no loss in generallty in assurnlng

that cn-r >

The following lemma will þe useiL to evaluate

,o,rlo^ l.-" L"n cn-1...C1J

Ï,emma

Def 1ne

cL = mln(max( ân r cn- " ) ,bn ) ( za)

ancl

ïlIn (2e)

(;o¡

L

zN, ¿

X=â

= ¿ "U/n

bn

â¡ Cn-t...C1 lbn

-F w.Tll¡ d. Cn-1orrG1

(

190.

Proof

Consid.er ttre foJ.J.owing 3 eases.

a) ch-r <

Frorn the d.ef initiorr of d. it foJ-lows that d' = ân'

As â¡>it follows that lf xn € [anrbn] attd

x¡-1 € ["n-rrxn]r..ex1 € lcrrxn]

then xn 2 max[*n- ",

. . .xt J

andhence Xl,t(*nr.."x1)=ïtr fromeQ.1. 3l)Hence from eq.25

b

l?rH

"n

â¡ Cn-to. rG1

Xn=d Xn-t=Cn-1 X1=c1

xH'l (*n, . . . x1 )d.xn. . .d.x1

ïyd.xn . o . dx1 f rom eq. J1 .

rr aaa

rXn=d'

n

IXn n

v-^

^D- 1-vft- 1 Â1-"1

I]n

t

tL

=W

X=Ô

(*-" r ) dx

n

tb

=ï/Wn d cn-1...c1

191.

And. as Wnd.

An Cn-1oorC1

_^ v -ô^n-an ^n-1-vn-1 ^1-v1

anit expand.ing this integral

- O because ân=d it follows

x\' ¿ (*n, .' .x1 ) dxn.' . d.x,

lthat eq.29 hoId.s in this câsê¡

(¡) cn,r€ [an,bn] 3z)From the d-efinition of d-'

d. = min(max(ân r cn- r) ,br,) and using eq.32

it follows that d. = min(cn-rrbn) = cn-r , 3l)

When xn € [ar,rcn-r] and xn-r € ["n-r rxn],

then xn-1 >

x1,¿(*nr...x1) = !, . (¡+)

Similarly, when xn € [cn-rrbn] and. xn-r € ["n-rxn],xn-z € ["n-r rxn] r. o .x1 € [c1 rxn], since in eq"2J cn-t

has been d.efined. in such a way that cn-r >

it follows that

xn>

and- thus XA,x (*n, o o.x1) = w from eq.1. (lS)

Hence from eq.25

b

lzH'xn

An Cn-1o. oC1

aai

n

/" r" aat

Xn=ân Xn-1=Cn-1 X.7=C,

xH, ¿ (*n, . . .x")d-xn . . .d.x,

r92,

+ i'" f" ..i f" xH,¿(*nr...xr.)d.x...d.x1llt'"Xn=d X¡-1=G¡*t Xt=cl

Now employi-ng tlre fact tlrat d. = cn-1 and. apþlylng eq.s.34

ar..d- 35 thls lntegral may be written

nþnAdxn...dx1

n

¡.ò flzI't ân Cn-1...C1

Xn=ân Xn-t=Cn-t X1=Ct

+

fXn =ân

d_

lWn an Cn-tonrCl

Hence eq"29 hold.s.

(") Gn-1 ) b.n

FirstNow lf

Xn =d. X¡- 1=C¡- L Xt=QLrn

+ ur l/ifn

wd.xn...d-X1¡aa

Ln

tñt (*n-. r ) ctxn + vu fo''= 1

*^luigi(xn-c, )dxn

lbn

d. Cn-1'..C1

lhis case is nather slmilar to

d = miïr.(*r*(ânrcn-r)rbn) = mln(cn-1rbn) = b¡'

xn € [ an ,bn ] and. xn- 1 G ["n-, ,xn ] ,

xn-a ê ["n-"rxn] rrorxl € latrxn],

(r).

then since cn- r Þ maxlcn -z'- .. . cl J, 1t follows

193.

that XN'¿(*nr...x1) = !'.

Hence from eq"ZJ, ca31.ying out the ealculatlon in tlre same

way as before it can be shoum that

/Wo

n

Furthermore bn=dn implies that TVn d cn-1crrC1

thus

z\,¿ lbn

an Cn-1...Cd_

an Cn-t...Qa I

n

An Cn-too.C1

d.

An Cn-foo.C1

b

l o

tb

+zl,x = [Wn

n lb

+ UrWn d Cn-1. o.C1

and. eq.50 hold.s.

This completes the proof of the Jemma.

Now eqs.28 JO may be useaL in coniunction with

eqs . 22 ,23 and 26 to. glve:

lbn b1

â¡.roâ1> p(s)zï,x

SeJ

n

an Cn- 1 . . r C1tb

xH '¿

where

(]0)

l /Wn l¿bn d

an Cn-1n..C1

lbnd cn-1c"oG1

ZY, an Gn-t.. oC1

+ urvfn

194.

d. = rnin(nax(ân rcn-t) ,bn)

w"L: ""-1cc.",-l

= I iil'(*-"')a*X=â

and. Sr,rrp(S) have þeen d.eflned. earJ.1er.

Exa'nPle F ¡" ba br -l

r,q.36 nay be used to ftnd xä'J L ;; "; ";l

,

Using eq.22 Let,

S1 = [.a, ,atl, Sp = [a"rb, l, Ss = lbrrarJ,S4 = [b, ,b, J,

then by the d.efinitlon glven earller J = [Sr.rSrrS"rS¿ l,and. uslng eq.23

p(Sr) = 1, p(sz) = -1 , p(ss) = 1, p(s¿) = -1 .

Also fron eq.29

Vl¡ lb

àQzC1 x-cfr(t=1

d.xt)

X=â

X=â

(x'- ( cr+c, ) x+er c, ) d.x

= t{t"-at ) -}(cr+cr¡ ltz-az ) +""c, (u-a)

Thus from eq.JO

r95.

zä'¿ lWsl l lb3

a3 c2 c1

d

âs Cz CL+ wlllg

bnd- cz cL

where fnom eq.28 d = nin(max( àascz) rb").Hence substltuting into eq,.36

xä ,¿ l :l l

l

b3 b2 b1

a9 ã2 aL - zU'¿bg

âs ã2 Aazl,¿

b3

âs azb

ßt)b3

âs bz brb3

â3 bz d1+ Zä'¿ t

A fortran subroutine TI¡as programmed. to evaluate

this funotlon.The integral Xl,

the same wâ$.

From eq.J6

may be evaluated. int b¿ brà.1 .otã.1

xl '¿ l lb¿ b1

Ð.1 ..oã'7 = Zf ,l

- zl'¿

zl"

âs ?2 A1

aL âs bz at

b4

ã1 bs ã2 ãa

b¿

ã1 bs þa aL

zr ,t

+ ZY.'t

b¿

&1 âs ez bt

b4

ã4 s's bz bl

b1

8¡, bs dz bt

b4

8,1

b,

ll l+ Zl'¿

t l l¿b4

a4 b3 bz br+ zl,,¿ zl,

196.

Now from eet2)

Íl,r l

+

ba'Cs C2 C1

fi (*-", )a*

(c"+cr+c" ) (¡t -a" )b4 -q1

34 +

(crcr+c, ca+cpca ) (¡=-a' )2

lfrr¡s using eesoJO and. 38,

zl,¿ = [W,lþ¿

AL Cg Cz Ct

d.

A4 Cg Cz C1 tb4

+ TtrlV¿ d. ca cz cl Iwith d. - rnin(max( ãs gez) ,b" ) .

A fortran function was written to evaluate thls

integral.

A.l+. Use of monte-carlo meth!¡d. to check valid-ity of the

algebraic expression

A stand.ard. monte-canlo method. has been used to

show that the analytic elcpressions obtained. for

Xä '¿ lb3 b2 blâs à2 àa

arrd. Nl'xbL b1

Lcctã.¡

are correct. .A.s the method. used. 1s a standard. onet

d.etalls of the procedure ï1111 not be given here but may be

for:rrd- if required. in (1il. Furthermore the rand.om gener-

t97.ator used- has been tested. and- found. to be satisfactory"

Details are given in append.ix B.

The table bel-ov¡ glves the results of an investiga-

tion where values ofvw./13 ¿

bs

âs

bz brã2 ar

were calculated. for iliffering values of al ,bt using the 2

method.s, The monte-carlo method. employed. 1OrO00 random

points (see (SÐ for e:çlanation), and. rlv TI/as set to +5

anil l. to -2o

Closer agreement between the 2 sets of figures 1s

obtalned when the number of random polnts used by the monte-

carlo method 1s lncreased. For example, 1n the worst poss-

lble case, the |E}il Ilne of the table shows that for 10,000

polnts the monte-carIo method glves a value of .0032 wh11e

0. 3333o.0l+17

-0. t2l+T0.00000.0000o.04150.0023

-o.0255-o.02670.0083

-0.0330

0.36980.0\29

-0.135?0.00000.00000.01+160.0032

-o,0255-o.02690.008?

-0.0325

I10000000o0

000000900Boo

500500900Boo?00800306

0.0000. 5000.5000.2000.0000.2000.2000.600o. l+oo

0.2000.100

III00000000

000000000T002002006oo5008oo6oo8oo

00000000000

0005005007002001005002006oo500300

11I0I000000

0000000009oo000?oo700?007ooT00\oo

00000000000

00050000100000l+oo

hoohoohoohoo200

(1)(z)(3)(h)( r)(6)(r)(B)(e)

( ro)(u)

AnalyticMethod

Monte-CarloMethotl.

bsâ3b2d2brê1

198.

repeating the cal-cuJ-ation usÍng 1Os points (and. taking

20 sec. of copc time) glves a value of .0024 whlch is

much nearer to the correct a¡rswen. Hoïr/ever this was not

d.one !f1th the remaining set of results for reasons of

economye Thus the results serve to confirm the accuracy

of the analytic expresslon given by eq.JJ"

Note that the monte-carlo method. could not itsel-fbe usecL to evaluate these integrals because it is¡

(") too slow (Z sêc. per 1O,OOO points)

(¡) not sufficiently accu::ate unless exceptlonal--

ly large numbers of points are us ed."

The valid.ity of the analytic ercpression obtalned. for

Nl 'tba b1

â4o."â1 lwas conflrmed. in the same ï/âgr

L99.

APPENDIX B

RANDOM NUMBER GENERATOR

Thls appendlx wl_Il d.lscuss the random number gener-

ator used to lmplement the simulatlon technlques employed

ln thls study.

The random number generator used 1s of the llnear

congruentlal type, see . Kn¡th (21), and 1s provlded as a

standard llbrary subroutlne (m¡e) on the CDC-6000 serles

conputers (see CDC Reference Manuat (5)).

The foIIowlng statlstlcal test was applled to thls

generator. Slnce thls ls a standard test (14r17r18121),

only a brlef outllne w111 be given here, 1n whlch a faml]l-

arlty wlth the detalls of thls test w111 be assumed.

The random number generator provlded a sequence of

numbers whlch r,tere tested for unlform dlstrlbutlon over the

lntervat (0r1) as follows. The interval was dlvlded lnto

ten equal sublntervals, and fr , the number of random

numbers falI1ng lnto. the 1-th lnterval was counted.

If n ls the length of the sequence, then the

statlstlc

10.n

10

=7.

2

xi no )

1

has a X2 dlstrlbutlon wlth 9 degrees of freedom. Thls

experlment ls repeated m tlmes and the m values of Xî

200.

thus obtalned are conpared wlth the theonetically expected

dlgtrlbutlon of Xl values. The nesultg obtalned wlth

n-1000 and m-1000 show that the assertlon thaf the

orlglnal rand,om numþer sequence l-s unlformly dlstrlbuted ls

cornect at the 60l level, a satlsfactory result (see

Sneclecor and."Cochran (¡l+) ).A second test was canrled out whene fr I wqs defln-

ed to be the numben of numbers ln the l-th lntenvalr fo11.-

owed by a numben ln the J-th lntervaL. The X2 statlgtlcthus becomes

x2= 10sn t r t=1

10 (fr r - 1þ)".

Proçeedlng as before, 1t was found that the assertlon that

congecuülve random numbers are not palnrlse conrelated, wag

comect at the 901 level, agaln a satlsfactory nesult.

Generatlon of n umbers fr s numb

The foLlowlng, standand, technlguêr 1s used to

generate 11, non-sorrelated random numbers from one sLngle

random numben. Thls method w111 be descrlbed for the case

n-2 from whlch the generalizaþLon of thls method forn > 2 ls an oþvlouo one,

The method works by dlvldlng the lntenvaL (0r1)

lnto m2 equal. sublntenvals Ir r...Ir2 such that theee

sublntervals cover the lnterval (0r1). A 1-1 mapplng,

201.

f , ls then d,eftned from any lnterval I¡ to some lntegen

p¿lr (1,J) wherc (r'J) beLongs to ühe set

g r {(or0).,.(0rh)r(rr0)...(1,n)...(nr0)...(nrn)} wlth rl=t¡¡-l.

thls mapping ls deflned ln the followlng way'

Supposethatkhastheunlquerepresentatlonk = m(l-I) + J whene o < J s m-1,

then f(Ir) maps on to (1,J).

Now deflne P¡ = # and tz = # ,

thenslnce Oslsm-l and O<

0sllrrl and 03'â2 <1. Thusthefunctlon f maybe

used to produee the two random numbers rl, t2 fnom the

s1ngLe rand.om number P. AIso, sLnce f ls a 1-1 mapplng'

the components of the nandom palr 1rl r9z are not correlated

1f the orlginal sequence ls not correlated'

The above method is used to genenate random numberE

whlch are used ln slmulatlng the hands dealt ln the z-PQ,

3-PG and A-PG (see chapters 4 and 5). In actual lmplement-

atlon the above process ls easlly carnled out by dlvldlng

the computer wolld lnto 2 halves, where P¡ 1e calculated

dlrectly from the flrst haLf of the wond, whlle t2 ls

calculated from the second ha1f.

202.

APPENDIX C

CALCULATION OF'PROBABILITIES

Thls appendlx descrlbes the use of a monte-carlo

method t.o calculate the probablIltles of wlnnlng assoelated

wlth an;y glven poker hand ( eee chapter 2) , Two types of

probabj.ll.iles are requlred-.

(a) f¡(r¡), the probablllty that a glven hand has

of beatlng any other random unlmproved hand'

(b) fa(h), the probablllty that a glven hand h

has of beatlng any nand'om lmproved hand'

Slnce the method.s used to flnd f¡(h) and fa(h) are very

slmllar, only the caleulatlon of f6(h) wllL be descrlbed

ln detall.The method used to caleulate ft(h) ls based on

the folloi,rrlng ldea.

Glven airy hand h, fb(h) I ñâV be approxlmately determlned

thus. F1rst, a very large nurnber, N, Of random hands

are generated. The methoC of hand genenatlon ls the sane

as was used for thé poker simulator (see ehapter 2). It

has been shown ln ohapter 2 that the hands produced 1n this

way are random. Next, L, the number of tlmes that the

glven hand h beats the randomly generAted hands, ând LDt

the nutnber of tlnes that the glven hand draws wlth the

random hands, 18 found. Then f¡(h) ls approxlmated by

(L+LD/2)/ñ. Slnce thls process ls tlme consumlng (for

203.

large N) Bome meaaures were taken to speed the caLculatlon.

1. Instead of generatlng N random hands each

tlme that fb(h) ls calculated, ühe set of

N random hands ls only generated once, and

ühen stored. All. fb(h) values ane then è

ealcuLated wlth respect to thls aane set of

random hands.

2, The pnocess of determlning whether one hand

beats another hand 1s repeated many tlmes

durlng the executlon of thls algonlthm.

Fon thls reason the fo1low1ng method was

devlsed to speed up thls hand comparlson

process.

ft 1s posslble to deflne a functlon q(h) (see

later) whlch asslgns a unlque lnteger to every hand h.

Furthenmore, thè functlon q(h) has the property that,

hand hr beats hand hz lf anrl only lf A(hr) > q(h2)' and

hand hr draws wlth hand hz 1f and only lf O(nr) = q(ha).

The deflnltlon of thls functlon, and the proof that 1t has

the above pnopertles, w111 be glven Later ln thls text.Thus, lnstead of storlrtg N nandom handsr the N

correspondlng numbers, âs deflned by the funetlon q(h),

are stored. Then, glven any hand h, q(h) rnay be ealcula-

ted for that hand, and the hand comparlson process may be

achleved by t'estlng the numþer of tlmes ühat the lnteger

204.

q(h) ls greater than on equal to the glven lntegers ln the

aet of N numbers already generated.

TÌre rnethod used to store the N numbers was to put

them ln an array (A(f), I=lrN). An alternatlve approach

would be to have an aruay C(I) , c(2)r.. "C(K) where the

elenent C(J) counts the nurnber of tlmes that the lnteger

J oecurs ln the set of N numbers. However, slnce

K(maxlnum of q(h)) ls appnoxlmately L37 (see deflnltlonof q(h) Iater 1n text), lt ls clear.ly lmpractlcal to

employ the latten method.

A drawback of the method used here ls that lt does

not glve a rellab1e estlmate of how often a particular type

on cl,ass of hand occurs when the percentage of ocCurnences

of that partlcular class among the N random hands gener-

ated ls small. But, for the purposes of thls study,

lnvar'lably only large classes of hands are dealt wtth, forexample, the percentage of hands wealter than a palr of 8rs.

Hence thls partLcular dlsadvantage, 1n thls cå.se, ls of no

practlcal consequence.

Íhe process of flndirlg fu(t¡) wlII now be summarls-

ed.

(1) Generate a nandom hand h, and caleuiated q(h).

Repeat thls process N tlmes (tl = 301000) and

store the resultant numbers 1n the arrey

(n(f ), I=1,N).

205,

(11) Glven any hand h for whlch fu(fr) ts requlredt

flrst ealculate q(h). Now compute h, the

number of tlmes that q(h) >

number of tlmes that q(h) = A(r), fon I-L,N'

Then fr(h) ls appnoxlmated by (L+LD/z)/N.

A veny slmltar process ls used' to compute fa(h)

except thls tlme the array (A(I), I=1,...N) ls computed

ln the followlng way. A random hand 1s dealt, and 1f thls

hand 1s a palr or better then 1t ls randomly lmproved,

taklng lnto account cards already held, to some new hand h{'

If the hand ls weaker than I palr lt ls dlscarded. Thls

selectlve process ls used because, ln pokef, players seLdom,

1f ever, play on less than a paln (see chapter 5) ' As

before thls process was repeated N tlmes (not countlng

hands weaker than t palr) and. each tlme q(fr¡È) was stored

ln the aruay (A(I) ,I=1r. . .N). Now fa(h) may be ealcula-

ted ln the same $ray as fr(n).

Deflnttlon of the functlon q(h).

The functLon q(h) ls calculated accofdlng to

table 20. Thls table w111 not be dlscussed ln detall

aS lt ls not of central lmportance to thls thesls. However'

by careful examlnatlon of thJ-s table the followlng polnts may

þe valldated.

TABLE 20

DEFII.JIT]ON OF I]AND .JEOUENC]NG FUNCT]ON2C6 .

Hantl Type

1) less than 1 pair

2) I pair

3) 2 pair

)+) 3of akind.

5) straight

6) tlr¡.str

T) t'.¡11 ¡ook

B) l+ of a kincl

9) routine

Definition of variab.l.es

i1 ,f2ri3,fl+,I! ar.e hand.

vaIl,..sf , in cescend,il,g

oraler of rnagni'"ud.e

Il = value of pair12,f3,I)+, are hanti val-uesof r-rrpaired carcls indescending order of mag-nitud.e

Il=value of higher pairl2=va1ue of lower pairI3=val-ue cf unpaired. carcl

fl = value of trebleI2rI3 unnatched. carcls in

descend.i.ng orclerII = val-ue of highest carcl(Ace counts as O if it isfirst card. in straight)

II ,I2,I3,I\,I5 are handvalues in descending oralerof rnagnitud.e

Il- = value of three of akind

f2 = value of two of a kind.

fI = vâlue of l+ of a kind.

Il- as for straight (seeabove )

Value of q(h)

q( h) -I1. f3a+I2,133+I? .132+I)+ .f 3 i+Ii

nax.vafue of q(n) < 12(l-3q+.,+130)= re(r¡s-r) ; r3t !

q(rr ) =r 36+tr. r33+I2 l-32+I3. 13+Il+max,val-ue or q(¡)< 136+12( f3s+Ì32+13+r).136+13s

o ¡¡¡=13 5+136+II .r32+f 2 .13+r3nax q(tr) < 13s+136+13q

q( rr)=r34+r:t+I3t*rr. r32+r2. r3+r3

nax q(tr) < 2.134+13s+r36

q(¡) = 2.134+t3s+I36+11

max q(n) < 13+2.1-34+13s+l-36

q(h)=11+2.134+13s+136+r1.134+r2.133+r\..:-3+r5

max q(ir) < t3+2,134+13s+2.136

q( h) =13+2.13b+13 5+2.136+rr. t3+r2

nax q(h) < r3+13s+2.134+13s+2.136

q(h) =11+rf 3+2

. 13 r+.t-I3 s+2

. 13 6+.L1

nax q(tr) < 2.13+133+2.13q+f3s+2.136

q( ¡) =e. f3+r3 3+2. 13 4+13 s+2. f3 6+r1

max q.(rr) < 3.13+t-33+2.13a+13s+2.136 < 137

5 The sum of the G.P. a+ar+...*arn is given by Sn =a( r-rn-l )

t_ r+

hand values are assigned in the foJ-loving way (irr.espective of suit)CAIìD 2 3l+ : 6l g 9ro J q K A

0 12 3I+ 5 6l g g10 11 12VA].UE

207 ,

I. Each unlque hand (írreepeetlve of sult) 81ves

a unlque lnteger q(h) ' If hr and hz are

2 dlfferent hands then q(hr) / q(hz)' and lf

hr and hz are ldentteal hands then q(hl) = q(h2)'

2. The value of q(h) for hands of dlfferlng typest

ls so calculated that glven hands hr and hz

where hr ls a stnonger type of hand than t:zt

then q(hr) > q(h2).

3. ff 2 hands h1 and hz are of the same type,

but hand hr beats hand ]nzt then q(hr) > q(hz)'

Now, by uslng polnts 1 to 4 above lt w111 be shown

that:hand h1 beats hand h2 lf and onIY 1f

q(hr ) > q(h2 ) ' and h1 draws wlth hz lf

and onlY 1f e(frt ) = q(hz ).

Íh1s statement w111 be proved 1n 4 steps.

(1-) If h1 beats hz then by LrZ and 3 above'

9(hr) > q(h2)

(11) rf q(hr) > q(hz) then bv 1. and (1) above,

hand hr beats hand }:z'

(1i1) If hand hr draws with hand hz then

q(hr ) = q(hz ) as from (i) and (11) above

q(hr) r q(h¿) and q(hr) > q(hz)' thus

tFor deflnltion of hand types see table l.

208t

9(hr) must equel q(h2).

(fv) If q(hr ) = q(hz ) then hands hr and hz

must draw bY 1 âbove.

Thus , th'e proof 1s comPlete.

209.

FIGUBE 1

FIGURE 2

FIGURE 3

FIGURE 4

FIGURE 5

F'TGURE 6

FIGURE 7

FTGURE 8

FIGURE 9

FIGURE TO

FTGURE 11

FIGURE L2

INDEX OF G S

DECK SHUFFLING ALGORITHM

ALGORTTHM TO DETERMINE WHÏCH

CARDS TO DISCARD DURTNG HAND

TMPROVEIIENT

INTERACTI\TE POKER SIMULATOR

FLOI{ DIAGRAM OF LAH ALGORITHM

FLOI^I DIAGRAM OF 2-PG

STRATEGIES FOR THE z-PG

z.PG STRATEGY F'I]NCTIONS

SUMMARY OF STRATEGY FUNCTION

DEFTNITIONS AND GAME FLOI¡T

DIAGRAM FOR 3-PG

GAIUE FLOW DTAGRAM F'OR THE 4-PO

AND DEFÍNITION OF STRATEGY

FUNCTIONS

EQUIVALENCE BETI/IEEN 3-PG AND 4-PG

l^ll{EN PLAYER 1 DECLINES TO PtAf

COMPARISON BET!üEEN ANALYTIC PAYOFF

T'T'NCTIONS FOR THE 3-PG, AND RESULTS

OBIAINED FROM SIMULATION

SOLUTION TO THE 4_PA EXPRESSED IN

TERIVTS OF POKER HANDS

AN EXAMPLE OF A SIMPLE NETI¡IORK

Page

27

30

44

67

79

B2

B4

110

l22

L27

133

140

FIGURE 13 149

210.

TABLE ITABIE 2

TABLE 3

TABLE 4

TABLE 5

TABLE 6

TABLE 7

TABLE 8

TABTE 9

TABLE 10

TABLE 11

TABLE 12

TABLE 13

TABLE 14

TABLE 15

TABLE 16

INDEX OF TABLES

CI,ASSTFYING POKER HANDS

EPSTEINIS TABLE OF POKER

PROBABILITIES

PROBABILTTIES OF WTNNING FOR

UNIMPROVED HANDS, CALCULATED

BY A MONTE-CARLO }'IETHOD

HAND NUI!tsERTNG FUNCTION Pr(h)

PROBABILITIES OF IÂIINNING FOR

TMPROVED HANDS, CALCULATED BY A

IUONTE-CARLO METHOD

AN EXAMPLE OF THE BETTING ALGORITHM

AN EXAMPLE OF THE LAH ALGORÏTHM

SUMMARY OF PLAYS FOR von NEUMANN|S

GAII4E

COMPARISON BETüIEEN ROSEN I s MODIFIED

METHOD AND THE LAT]Ï ALGORTTHM

SUMMARY OF PLAY FOR T:IE z-PA

SOLUTION TO THE z-PG

SIMULATION OF z-PG PAYOFF FUNCTIONS

SUMMARY OF PLAY FOR TTIE 3-PG

SUMMARY OF PLAY IiOR THE 4-PG

SOLUTION TO THE 4-PG

EQUIVALENCE BETI,rIEEN HANDS IN THE

4-PG AND REAL POKER HANDS

Page

23

31

34

35

3B

41

56

7o

73

B7

94

103

115

L23

1,29

L37

21J.

TABI,E L7

TABI,E 18

TABLE 19

TABLE 20

MINIMUM HAND }'T¡TEN F.TRST TO PLAY:

THEONEEICAL RESULT

NTINIMUM HAl,lD ÏÍHEN FIRSI TO PLAY:

PRACTICAL CRITERIA

RESULÍS FOR NETVüORK OPTIMIZATION

DEFINITION OF HAND SEQUENCING

F'T'NCTION

Page

t42

145

154

206

2r2.

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6